2-Equations and inequations

# 2-Equations and inequations ## Interval notations, inequality signs and the number line, Linear inequalities in one unknown ### 1. Introduction Welcome to this lesson on inequalities! In mathematics, we often deal with quantities that are not equal but have a specific relationship. Inequalities help us describe these relationships, indicating when one quantity is greater than, less than, greater than or equal to, or less than or equal to another. This lesson for Form Four students will introduce you to various ways of representing these relationships, including interval notations and the number line, and equip you with the skills to solve linear inequalities in one unknown. ### 2. Key Concepts #### 2.1 Interval Notations Interval notation is a concise way to represent sets of real numbers. It describes a range of numbers between two endpoints. ##### 2.1.1 Open Interval An open interval includes all real numbers *between* two specified endpoints but does *not* include the endpoints themselves. * **Notation:** \( (a, b) \) * **Inequality:** \( a < x < b \) * **Number Line Representation:** An open circle (or unshaded circle) at \( a \) and \( b \), with the line segment between them shaded. ##### 2.1.2 Closed Interval A closed interval includes all real numbers *between* two specified endpoints and *also includes* the endpoints themselves. * **Notation:** \( [a, b] \) * **Inequality:** \( a \le x \le b \) * **Number Line Representation:** A closed circle (or shaded circle) at \( a \) and \( b \), with the line segment between them shaded. ##### 2.1.3 Half-Open / Half-Closed Intervals These intervals include one endpoint but not the other. * **Notation:** * \( [a, b) \): Includes \( a \) but not \( b \). Inequality: \( a \le x < b \). * \( (a, b] \): Includes \( b \) but not \( a \). Inequality: \( a < x \le b \). * **Number Line Representation:** A mixed representation with one closed circle and one open circle at the respective endpoints, and the line segment between them shaded. ##### 2.1.4 Unbounded Intervals These intervals extend infinitely in one or both directions. * **Notation and Inequality:** * \( (a, \infty) \): \( x > a \) * \( [a, \infty) \): \( x \ge a \) * \( (-\infty, b) \): \( x < b \) * \( (-\infty, b] \): \( x \le b \) * \( (-\infty, \infty) \): All real numbers. * **Number Line Representation:** An open or closed circle at the finite endpoint, with an arrow indicating the direction of infinity. #### 2.2 Inequality Signs and the Number Line ##### 2.2.1 Understanding Inequality Signs Here are the common inequality signs and their meanings: * \( < \): Less than (e.g., \( x < 5 \) means \( x \) is any number smaller than 5) * \( > \): Greater than (e.g., \( x > 5 \) means \( x \) is any number larger than 5) * \( \le \): Less than or equal to (e.g., \( x \le 5 \) means \( x \) is any number smaller than or equal to 5) * \( \ge \): Greater than or equal to (e.g., \( x \ge 5 \) means \( x \) is any number larger than or equal to 5) * \( \ne \): Not equal to (e.g., \( x \ne 5 \) means \( x \) is any number except 5) ##### 2.2.2 Representing Inequalities on a Number Line Representing inequalities on a number line provides a visual understanding of the solution set. * **Open Circle (\( \circ \)):** Used for strict inequalities (\( < \) or \( > \)), indicating that the endpoint is *not* included in the solution. * **Closed Circle (\( \bullet \)):** Used for inclusive inequalities (\( \le \) or \( \ge \)), indicating that the endpoint *is* included in the solution. * **Shading:** The portion of the number line corresponding to the solution set is shaded. An arrow indicates that the solution extends indefinitely. #### 2.3 Linear Inequalities in One Unknown ##### 2.3.1 Definition A linear inequality in one unknown is an inequality that can be written in one of the following forms: \[ ax + b < c \] \[ ax + b > c \] \[ ax + b \le c \] \[ ax + b \ge c \] where \( a, b, c \) are real numbers and \( a \ne 0 \). The goal is to find the range of values for the variable (the unknown, usually \( x \)) that satisfies the inequality. ##### 2.3.2 Solving Linear Inequalities Solving linear inequalities is very similar to solving linear equations, with one crucial difference regarding multiplication or division by negative numbers. ##### 2.3.3 Properties of Inequalities * **Addition/Subtraction Property:** You can add or subtract the same number from both sides of an inequality without changing the direction of the inequality sign. * If \( a < b \), then \( a + c < b + c \) and \( a - c < b - c \). * **Multiplication/Division Property (by a Positive Number):** You can multiply or divide both sides of an inequality by the same *positive* number without changing the direction of the inequality sign. * If \( a < b \) and \( c > 0 \), then \( ac < bc \) and \( \frac{a}{c} < \frac{b}{c} \). * **Multiplication/Division Property (by a Negative Number):** If you multiply or divide both sides of an inequality by the same *negative* number, you **MUST reverse the direction of the inequality sign**. * If \( a < b \) and \( c < 0 \), then \( ac > bc \) and \( \frac{a}{c} > \frac{b}{c} \). This is a very important rule to remember! ### 3. Vocabulary * **Inequality:** A mathematical statement that compares two expressions using an inequality sign (e.g., \( <, >, \le, \ge, \ne \)). * **Interval Notation:** A compact way to describe a set of real numbers between two endpoints using parentheses or brackets. * **Open Interval:** An interval that does not include its endpoints, represented by parentheses (e.g., \( (a, b) \)). * **Closed Interval:** An interval that includes its endpoints, represented by square brackets (e.g., \( [a, b] \)). * **Unbounded Interval:** An interval that extends infinitely in one or both directions, using \( \infty \) or \( -\infty \). * **Number Line:** A visual representation of real numbers, used to graph inequalities. * **Linear Inequality:** An inequality in which the highest power of the variable is 1. * **Solution Set:** The set of all values for the variable that make the inequality true. ### 4. Examples **Example 1:** Represent the inequality \( -4 < x \le 2 \) using interval notation and on a number line. * **Interval Notation:** \( (-4, 2] \) * **Number Line:** An open circle at -4, a closed circle at 2, with the line segment between them shaded. **Example 2:** Write the inequality and interval notation for the number line representation showing an open circle at 3 and shading extending to the left. * **Inequality:** \( x < 3 \) * **Interval Notation:** \( (-\infty, 3) \) **Example 3:** Solve the inequality \( x + 7 \ge 10 \). Represent the solution as an inequality, in interval notation, and on a number line. * **Solution:** \[ x + 7 \ge 10 \] Subtract 7 from both sides: \[ x \ge 10 - 7 \] \[ x \ge 3 \] * **Interval Notation:** \( [3, \infty) \) * **Number Line:** A closed circle at 3, with shading extending to the right. **Example 4:** Solve the inequality \( 3x - 5 < 13 \). Represent the solution in all three forms. * **Solution:** \[ 3x - 5 < 13 \] Add 5 to both sides: \[ 3x < 13 + 5 \] \[ 3x < 18 \] Divide by 3 (positive number, sign remains the same): \[ \frac{3x}{3} < \frac{18}{3} \] \[ x < 6 \] * **Interval Notation:** \( (-\infty, 6) \) * **Number Line:** An open circle at 6, with shading extending to the left. **Example 5:** Solve the inequality \( 1 - 2x \ge 9 \). Represent the solution in all three forms. * **Solution:** \[ 1 - 2x \ge 9 \] Subtract 1 from both sides: \[ -2x \ge 9 - 1 \] \[ -2x \ge 8 \] Divide by -2 (negative number, **reverse the inequality sign**): \[ \frac{-2x}{-2} \le \frac{8}{-2} \] \[ x \le -4 \] * **Interval Notation:** \( (-\infty, -4] \) * **Number Line:** A closed circle at -4, with shading extending to the left. **Example 6:** Solve \( 4(x - 3) > 2x + 6 \). Represent the solution in all three forms. * **Solution:** \[ 4(x - 3) > 2x + 6 \] Distribute 4: \[ 4x - 12 > 2x + 6 \] Subtract \( 2x \) from both sides: \[ 4x - 2x - 12 > 6 \] \[ 2x - 12 > 6 \] Add 12 to both sides: \[ 2x > 6 + 12 \] \[ 2x > 18 \] Divide by 2: \[ x > 9 \] * **Interval Notation:** \( (9, \infty) \) * **Number Line:** An open circle at 9, with shading extending to the right. ### 5. Exercises Solve each inequality and represent its solution in three ways: as an inequality, in interval notation, and on a number line. 1. \( x - 8 < 15 \) 2. \( 2x + 3 \ge 11 \) 3. \( 5 - x > 2 \) 4. \( \frac{x}{4} + 6 \le 7 \) 5. \( -3x + 4 < 16 \) 6. \( 2(y + 5) \ge 3y - 1 \) 7. \( \frac{2x - 1}{3} < 5 \) ### 6. Assignment 1. Solve the inequality \( 5(x - 1) - 2x \le 4(x + 2) + 7 \) and express the solution in interval notation. 2. A taxi charges a flat fee of 50 shillings plus 30 shillings per kilometer. If a passenger wants to spend at most 200 shillings on a ride, what is the maximum distance they can travel? 3. The sum of three consecutive integers is at least 60. What is the smallest possible value for the first integer? 4. Represent the following situations using inequalities, interval notation, and on a number line: * The speed limit on a certain road is 100 km/h. * To be eligible for a scholarship, a student must have an average score of at least 75%. 5. A company's monthly profit (P) in thousands of shillings is given by \( P = 1500x - 3000 \), where \( x \) is the number of items sold in thousands. If the company wants to make a profit of at least 4500 thousand shillings, how many items must they sell? ### 7. Summary In this lesson, you've learned about the fundamental concepts of inequalities. You can now: * Express ranges of numbers using **interval notation** (open, closed, half-open, unbounded). * Understand and use various **inequality signs** (\( <, >, \le, \ge \)). * Visually represent inequalities on a **number line** using open and closed circles and shading. * Solve **linear inequalities in one unknown** by applying inverse operations. * Remember the critical rule: when multiplying or dividing both sides of an inequality by a **negative number, you must reverse the inequality sign**. Mastering these concepts is crucial for further studies in algebra and other areas of mathematics. Keep practicing to reinforce your understanding! ## Solutions to absolute inequalities and inequalities ### Introduction Welcome to this lesson on inequalities and absolute inequalities! In mathematics, equations help us find specific values, but inequalities allow us to describe a *range* of values. This is incredibly useful for modeling real-world situations, such as budget constraints, speed limits, or error margins. We'll start by revisiting basic inequalities, then move on to compound inequalities, and finally tackle absolute value inequalities, which deal with the distance of a number from zero. ### Key Concepts #### What are Inequalities? An inequality is a mathematical statement that compares two expressions using an inequality symbol. Unlike equations that use an equals sign \( = \), inequalities use signs that indicate a relationship of being greater than, less than, or equal to. * **Inequality Symbols:** * \( < \): less than * \( > \): greater than * \( \le \): less than or equal to * \( \ge \): greater than or equal to * **Solving Linear Inequalities:** Solving linear inequalities is very similar to solving linear equations, with one crucial difference: * When you multiply or divide both sides of an inequality by a **negative** number, you must **reverse the direction of the inequality sign**. Example: Solve \( -2x + 5 < 11 \) 1. Subtract 5 from both sides: \( -2x < 6 \) 2. Divide by -2 (and reverse the sign): \( x > -3 \) * **Graphing Inequalities on a Number Line:** * Use an **open circle** ( \( \circ \) ) for \( < \) or \( > \) to indicate the endpoint is *not* included. * Use a **closed circle** ( \( \bullet \) ) for \( \le \) or \( \ge \) to indicate the endpoint *is* included. * Shade the part of the number line that represents the solution set. #### Compound Inequalities Compound inequalities combine two or more inequalities. They are typically joined by the words "and" or "or". * ** ## of the form ### 1. Introduction Welcome to this lesson on equations and inequations! In Form Four mathematics, you will encounter various types of algebraic expressions. This lesson focuses specifically on **quadratic equations** and **quadratic inequations**, which are fundamental concepts with wide applications in science, engineering, and everyday problem-solving. We will learn how to identify them, understand their properties, and master the techniques to solve them systematically. By the end of this lesson, you should be able to solve quadratic equations using different methods and solve quadratic inequations by analyzing critical points and regions. ### 2. Key Concepts #### 2.1 What are Equations? An **equation** is a mathematical statement that asserts the equality of two expressions. It typically contains an equals sign (`=`). The goal when solving an equation is to find the value(s) of the unknown variable(s) that make the statement true. ##### 2.1.1 Quadratic Equations A quadratic equation is a polynomial equation of the second degree. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \] where `a`, `b`, and `c` are real numbers, and `a \ne 0`. The variable `x` is the unknown. #### 2.1.2 Methods for Solving Quadratic Equations There are several methods to solve quadratic equations: * **Factoring:** This method involves expressing the quadratic expression `\( ax^2 + bx + c \)` as a product of two linear factors. If `\( (px+q)(rx+s) = 0 \)`, then either `\( px+q = 0 \)` or `\( rx+s = 0 \)`. * **Completing the Square:** This method involves manipulating the equation to form a perfect square trinomial on one side, allowing you to take the square root of both sides. * **Quadratic Formula:** This is a general formula that can be used to solve any quadratic equation. For `\( ax^2 + bx + c = 0 \)`, the solutions for `x` are given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] ##### 2.1.3 The Discriminant The expression `\( b^2 - 4ac \)` from the quadratic formula is called the **discriminant**, often denoted by `\( \Delta \)` (Delta). It helps determine the nature of the roots (solutions) of a quadratic equation without actually solving for them. * If `\( \Delta > 0 \)`, there are two distinct real roots. * If `\( \Delta = 0 \)`, there is exactly one real root (a repeated root). * If `\( \Delta < 0 \)`, there are no real roots (two complex conjugate roots). #### 2.2 What are Inequations (Inequalities)? An **inequation** (or **inequality**) is a mathematical statement comparing two expressions using an inequality symbol. Unlike equations, which usually have discrete solutions, inequations often have a range of solutions. ##### 2.2.1 Inequality Symbols * `\( > \)`: greater than * `\( < \)`: less than * `\( \ge \)`: greater than or equal to * `\( \le \)`: less than or equal to ##### 2.2.2 Quadratic Inequations A quadratic inequation is an inequality involving a quadratic expression. Its standard forms are: * `\( ax^2 + bx + c > 0 \) * `\( ax^2 + bx + c < 0 \) * `\( ax^2 + bx + c \ge 0 \) * `\( ax^2 + bx + c \le 0 \) ##### 2.2.3 Methods for Solving Quadratic Inequations Solving quadratic inequations involves finding the range of `x` values for which the inequality holds true. This is often done using: * **Graphical Method:** Sketch the parabola `\( y = ax^2 + bx + c \)`. The solution to `\( ax^2 + bx + c > 0 \)` corresponds to the `x`-values where the parabola is above the x-axis, and `\( ax^2 + bx + c < 0 \)` corresponds to where it is below the x-axis. * **Sign Table / Test Point Method:** 1. Rearrange the inequation so that one side is zero (e.g., `\( ax^2 + bx + c \)`) and the other side is `0`. 2. Find the **critical points** by treating the inequation as an equation and solving for `x` (i.e., find the roots of `\( ax^2 + bx + c = 0 \)`). These points divide the number line into intervals. 3. Test a value from each interval in the original inequation to determine if the inequality holds true for that interval. 4. Write the solution using interval notation or set-builder notation. ### 3. Vocabulary * **Equation:** A mathematical statement asserting the equality of two expressions. * **Inequation (Inequality):** A mathematical statement comparing two expressions using an inequality symbol (`\( < \)`, `\( > \)`, `\( \le \)`, `\( \ge \)`). * **Quadratic:** Pertaining to a polynomial of degree two. * **Coefficient:** A numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g., `a`, `b` in `\( ax^2+bx+c \)`). * **Constant Term:** A term in an algebraic expression that has a fixed value and does not contain any variables (e.g., `c` in `\( ax^2+bx+c \)`). * **Root (Solution):** The value(s) of the variable that satisfy an equation or inequation. * **Discriminant:** The expression `\( b^2 - 4ac \)` in the quadratic formula, which determines the nature of the roots. * **Critical Points:** The roots of the corresponding quadratic equation `\( ax^2 + bx + c = 0 \)`, which divide the number line into intervals for solving inequations. * **Interval Notation:** A way of writing subsets of the real number line, often using parentheses `\( ( ) \)` for strict inequalities and brackets `\( [ ] \)` for inclusive inequalities (e.g., `\( (2, 5] \)` means `\( 2 < x \le 5 \)`). ### 4. Examples #### Example 1: Solving a Quadratic Equation by Factoring Solve `\( x^2 - 5x + 6 = 0 \)`. **Solution:** We need two numbers that multiply to `6` and add to ` -5 `. These are ` -2 ` and ` -3 `. `\( (x - 2)(x - 3) = 0 \)` Thus, `\( x - 2 = 0 \)` or `\( x - 3 = 0 \)`. `\( x = 2 \)` or `\( x = 3 \)`. #### Example 2: Solving a Quadratic Equation using the Quadratic Formula Solve `\( 2x^2 + 3x - 5 = 0 \)`. **Solution:** Here, `\( a = 2 \)`, `\( b = 3 \)`, `\( c = -5 \)`. Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] \[ x = \frac{-(3) \pm \sqrt{(3)^2 - 4(2)(-5)}}{2(2)} \] \[ x = \frac{-3 \pm \sqrt{9 + 40}}{4} \] \[ x = \frac{-3 \pm \sqrt{49}}{4} \] \[ x = \frac{-3 \pm 7}{4} \] So, `\( x_1 = \frac{-3 + 7}{4} = \frac{4}{4} = 1 \)` And `\( x_2 = \frac{-3 - 7}{4} = \frac{-10}{4} = -\frac{5}{2} \)` The solutions are `\( x = 1 \)` and `\( x = -\frac{5}{2} \)`. #### Example 3: Solving a Quadratic Inequation Solve `\( x^2 - x - 6 > 0 \)`. **Solution:** 1. **Find critical points:** Set `\( x^2 - x - 6 = 0 \)`. Factoring gives `\( (x - 3)(x + 2) = 0 \)`. So, `\( x = 3 \)` or `\( x = -2 \)`. These are the critical points. 2. **Test intervals:** The critical points divide the number line into three intervals: `\( (-\infty, -2) \)`, `\( (-2, 3) \)`, and `\( (3, \infty) \)`. * **Test `\( x = -3 \)` (in `\( (-\infty, -2) \)`):** `\( (-3)^2 - (-3) - 6 = 9 + 3 - 6 = 6 \)`. Since `\( 6 > 0 \)`, this interval satisfies the inequation. * **Test `\( x = 0 \)` (in `\( (-2, 3) \)`):** `\( (0)^2 - (0) - 6 = -6 \)`. Since `\( -6 \not> 0 \)`, this interval does not satisfy the inequation. * **Test `\( x = 4 \)` (in `\( (3, \infty) \)`):** `\( (4)^2 - (4) - 6 = 16 - 4 - 6 = 6 \)`. Since `\( 6 > 0 \)`, this interval satisfies the inequation. 3. **Write solution:** The solution is `\( x < -2 \)` or `\( x > 3 \)`. In interval notation: `\( (-\infty, -2) \cup (3, \infty) \)`. ### 5. Exercises Solve the following equations and inequations: 1. `\( x^2 + 7x + 10 = 0 \)` 2. `\( 3x^2 - 11x - 4 = 0 \)` (Use the quadratic formula) 3. `\( x^2 - 4x + 4 = 0 \)` 4. `\( x^2 - 9 < 0 \)` 5. `\( 2x^2 + 5x - 3 \ge 0 \)` ### 6. Assignment 1. Solve for `x` in `\( 4x^2 - 12x + 9 = 0 \)`. What is the nature of its roots? 2. Find the values of `k` for which the equation `\( x^2 + kx + 4 = 0 \)` has two distinct real roots. 3. Solve the inequation `\( 3x^2 - 7x + 2 \le 0 \)` and express your answer in interval notation. 4. A rectangular field has an area of `\( 60 \)` square meters. Its length is `\( 4 \)` meters more than its width. Find the dimensions of the field. 5. Solve `\( (x+1)(x-5) \ge 7 \)`. ### 7. Summary In this lesson, we explored the fundamental concepts of quadratic equations and inequations. We learned that quadratic equations are of the form `\( ax^2 + bx + c = 0 \)` and can be solved using factoring, completing the square, or the quadratic formula `\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)`. The discriminant `\( \Delta = b^2 - 4ac \)` helps determine the nature of the roots. For quadratic inequations like `\( ax^2 + bx + c > 0 \)`, we learned to find critical points by solving the corresponding equation, then testing intervals on a number line to determine where the inequality holds true. Solutions to inequations are typically expressed as intervals or sets of numbers. Mastering these techniques is crucial for further studies in mathematics and various scientific fields. ## Solutions to quadratic inequalities. ### Introduction Welcome to this lesson on quadratic inequalities! In previous lessons, you learned how to solve quadratic equations, which involve finding specific values of \( x \) that make an equation true. Quadratic inequalities, on the other hand, require us to find a *range* of values for \( x \) that satisfy an inequality. This topic is crucial for understanding more advanced mathematical concepts and for solving real-world problems involving optimization and constraints. ### Key Concepts #### What is a Quadratic Inequality? A quadratic inequality is an inequality that can be written in one of the following standard forms: * \( ax^2 + bx + c > 0 \) * \( ax^2 + bx + c < 0 \) * \( ax^2 + bx + c \ge 0 \) * \( ax^2 + bx + c \le 0 \) Where \( a \), \( b \), and \( c \) are real numbers and \( a \ne 0 \). The goal is to find the set of all real numbers \( x \) that satisfy the inequality. #### Methods for Solving Quadratic Inequalities There are two primary methods for solving quadratic inequalities: ##### 1. Graphical Method This method involves sketching the graph of the related quadratic function \( y = ax^2 + bx + c \) and determining where the graph lies above or below the x-axis. **Steps:** 1. **Rewrite the inequality** so that one side is zero (e.g., \( ax^2 + bx + c > 0 \)). 2. **Find the roots (x-intercepts)** of the corresponding quadratic equation \( ax^2 + bx + c = 0 \). These roots are also called **critical values**. * You can use factoring, the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] or completing the square. 3. **Sketch the parabola** \( y = ax^2 + bx + c \). * If \( a > 0 \), the parabola opens upwards. * If \( a < 0 \), the parabola opens downwards. * Plot the x-intercepts found in step 2. 4. **Identify the region** of the graph that satisfies the inequality: * For \( > 0 \) or \( \ge 0 \), look for the parts of the graph above or on the x-axis. * For \( < 0 \) or \( \le 0 \), look for the parts of the graph below or on the x-axis. 5. **Write the solution set** using interval notation or inequality notation. ##### 2. Algebraic Method (Sign Analysis) This method uses the critical values to divide the number line into intervals and tests the sign of the quadratic expression in each interval. **Steps:** 1. **Rewrite the inequality** so that one side is zero (e.g., \( ax^2 + bx + c > 0 \)). 2. **Find the critical values** (roots) of the corresponding quadratic equation \( ax^2 + bx + c = 0 \). 3. **Plot the critical values** on a number line. These values divide the number line into two or three intervals. 4. **Choose a test point** within each interval. 5. **Substitute each test point** into the original inequality (or factored expression) to determine if the inequality is true or false for that interval. The sign of the quadratic expression (positive or negative) is consistent throughout each interval. 6. **Identify the intervals** that satisfy the inequality. 7. **Write the solution set** using interval notation or inequality notation. * Remember to include the critical values in the solution if the inequality includes \( \ge \) or \( \le \). #### Special Cases for Roots (Discriminant \( \Delta = b^2 - 4ac \)) * If \( \Delta > 0 \): Two distinct real roots. The parabola crosses the x-axis at two points. * If \( \Delta = 0 \): One real root (a repeated root). The parabola touches the x-axis at exactly one point (its vertex). * If \( \Delta < 0 \): No real roots. The parabola does not intersect the x-axis. It is either entirely above the x-axis (if \( a > 0 \)) or entirely below (if \( a < 0 \)). ### Vocabulary * **Quadratic Inequality:** An inequality involving a quadratic expression. * **Critical Values:** The roots of the corresponding quadratic equation, which divide the number line into intervals. * **Parabola:** The U-shaped graph of a quadratic function. * **Roots (x-intercepts):** The points where the graph of a quadratic function crosses the x-axis; values of \( x \) for which \( y=0 \). * **Discriminant:** The expression \( b^2 - 4ac \) from the quadratic formula, which determines the nature of the roots. * **Interval Notation:** A way to represent solution sets using parentheses and brackets (e.g., \( (2,3) \) or \( [-1, 5] \)). ### Examples #### Example 1: Solving \( x^2 - x - 6 > 0 \) (Graphical Method) 1. **Related equation:** \( x^2 - x - 6 = 0 \) 2. **Find roots:** \( (x-3)(x+2) = 0 \). Roots are \( x=3 \) and \( x=-2 \). 3. **Sketch parabola:** Since \( a=1 > 0 \), the parabola opens upwards. It crosses the x-axis at \( x=-2 \) and \( x=3 \). 4. **Identify region:** We want \( x^2 - x - 6 > 0 \), which means the part of the graph *above* the x-axis. ``` ^ y | | -------|-------------> x -2 | 3 | / |/ ``` (Imagine a parabola opening upwards, passing through -2 and 3.) The parts of the parabola above the x-axis are to the left of \( x=-2 \) and to the right of \( x=3 \). 5. **Solution:** \( x < -2 \) or \( x > 3 \). In interval notation: \( (-\infty, -2) \cup (3, \infty) \). #### Example 2: Solving \( 2x^2 + 5x - 3 \le 0 \) (Algebraic Method) 1. **Related equation:** \( 2x^2 + 5x - 3 = 0 \) 2. **Find critical values:** Factor the quadratic expression: \( (2x-1)(x+3) = 0 \) So, \( 2x-1=0 \Rightarrow x = \frac{1}{2} \) and \( x+3=0 \Rightarrow x = -3 \). The critical values are \( x = -3 \) and \( x = \frac{1}{2} \). 3. **Plot on number line:** ``` <-----|-------|-----> -3 1/2 ``` This divides the number line into three intervals: \( (-\infty, -3) \), \( (-3, \frac{1}{2}) \), and \( (\frac{1}{2}, \infty) \). 4. **Choose test points:** * For \( (-\infty, -3) \), let's use \( x=-4 \). * For \( (-3, \frac{1}{2}) \), let's use \( x=0 \). * For \( (\frac{1}{2}, \infty) \), let's use \( x=1 \). 5. **Substitute into \( 2x^2 + 5x - 3 \le 0 \) or \( (2x-1)(x+3) \le 0 \):** * **Test \( x=-4 \):** \( (2(-4)-1)(-4+3) = (-9)(-1) = 9 \). Since \( 9 \not\le 0 \), this interval does not satisfy the inequality. * **Test \( x=0 \):** \( (2(0)-1)(0+3) = (-1)(3) = -3 \). Since \( -3 \le 0 \), this interval satisfies the inequality. * **Test \( x=1 \):** \( (2(1)-1)(1+3) = (1)(4) = 4 \). Since \( 4 \not\le 0 \), this interval does not satisfy the inequality. 6. **Identify satisfying intervals:** The interval \( (-3, \frac{1}{2}) \) works. Since the original inequality includes \( \le \), the critical values themselves are included. 7. **Solution:** \( -3 \le x \le \frac{1}{2} \). In interval notation: \( [-3, \frac{1}{2}] \). #### Example 3: Solving \( x^2 + 2x + 5 < 0 \) 1. **Related equation:** \( x^2 + 2x + 5 = 0 \) 2. **Find discriminant:** \( \Delta = b^2 - 4ac = (2)^2 - 4(1)(5) = 4 - 20 = -16 \). 3. Since \( \Delta < 0 \), there are no real roots. The parabola does not intersect the x-axis. 4. **Determine opening direction:** Since \( a=1 > 0 \), the parabola opens upwards. 5. **Conclusion:** A parabola that opens upwards and never crosses the x-axis is always above the x-axis. This means \( x^2 + 2x + 5 \) is always positive (\( > 0 \)). Therefore, there are no values of \( x \) for which \( x^2 + 2x + 5 < 0 \). 6. **Solution:** No real solution (or the empty set, \( \emptyset \)). ### Exercises Solve the following quadratic inequalities. Express your answers using interval notation. 1. \( x^2 - 4x + 3 < 0 \) 2. \( x^2 + 6x + 8 \ge 0 \) 3. \( 3x^2 - 7x + 2 > 0 \) 4. \( -x^2 + 2x + 8 \le 0 \) ### Assignment Continue practicing with these problems. Show all your steps. 1. Solve \( x^2 - 10x + 25 > 0 \) 2. Solve \( 4x^2 - 9 \ge 0 \) 3. Find the values of \( x \) for which the function \( f(x) = x^2 - 5x + 7 \) is always positive. 4. Determine the solution set for \( (x+1)(x-4) \le 0 \). ### Summary Today, we learned how to solve quadratic inequalities using both the graphical and algebraic (sign analysis) methods. The key steps involve finding the critical values (roots of the corresponding quadratic equation), which help define the regions where the inequality holds true. Remember to consider the direction of the parabola (based on the sign of 'a') for the graphical method and to test points in intervals for the algebraic method. Special attention should be paid to the type of inequality symbol (\( <, >, \le, \ge \)) as it determines whether the critical values are included in the solution set. Mastery of these techniques is essential for future mathematical studies. ## Graphical solutions to linear simultaneous equations ### Introduction Welcome to this lesson on solving linear simultaneous equations using graphs! In previous lessons, you learned how to solve these equations using algebraic methods like substitution and elimination. Today, we'll explore a visual method that helps us understand *why* the solution works. By plotting the lines represented by each equation, we can find the point where they cross, which is the common solution to both equations. ### Key Concepts #### Understanding Linear Simultaneous Equations ##### What are Linear Equations? A linear equation is an equation that, when plotted on a graph, forms a straight line. It typically involves two variables, usually \( x \) and \( y \), and the highest power of these variables is 1. The general form of a linear equation is \( Ax + By = C \), or often, it's rearranged into the gradient-intercept form: \[ y = mx + c \] Where: * \( m \) is the gradient (slope) of the line. * \( c \) is the y-intercept (the point where the line crosses the y-axis). ##### What are Simultaneous Equations? Simultaneous equations are a set of two or more equations that share the same variables, and we are looking for values of these variables that satisfy *all* the equations at the same time. For linear simultaneous equations, this means finding a single pair of \( (x, y) \) values that makes both equations true. #### Graphical Solution Method The graphical method involves plotting each linear equation on the same coordinate plane. The point where the two lines intersect (cross each other) represents the solution to the simultaneous equations. This intersection point's \( x \) and \( y \) coordinates are the values that satisfy both equations. ##### Steps to Solve Linear Simultaneous Equations Graphically: 1. **Rewrite Each Equation in Gradient-Intercept Form (\( y = mx + c \)):** This makes it easier to create a table of values or identify the gradient and y-intercept for plotting. 2. **Create a Table of Values for Each Equation:** For each equation, choose at least three values for \( x \) (e.g., -2, 0, 2) and calculate the corresponding \( y \) values. Three points are better than two to ensure accuracy and catch any plotting errors. 3. **Draw the Coordinate Plane and Plot the Points:** Use a ruler and a sharp pencil. Label your axes (x-axis and y-axis) and choose an appropriate scale for both axes. Plot the points you found in your table for each equation. 4. **Draw the Straight Lines:** Carefully draw a straight line through the plotted points for each equation. Extend the lines so they intersect clearly. 5. **Identify the Point of Intersection:** Locate the point where the two lines cross. Read the \( x \) and \( y \) coordinates of this intersection point. 6. **Verify the Solution:** Substitute the \( x \) and \( y \) values from the intersection point back into *both* original equations. If both equations are satisfied, your solution is correct. ### Vocabulary * **Linear Equation:** An equation whose graph is a straight line. * **Simultaneous Equations:** A set of equations for which we seek a common solution (values of variables that satisfy all equations). * **Gradient (m):** The measure of the steepness of a line; change in \( y \) divided by change in \( x \). * **Y-intercept (c):** The point where a line crosses the y-axis (when \( x=0 \)). * **X-intercept:** The point where a line crosses the x-axis (when \( y=0 \)). * **Coordinate Plane:** A two-dimensional plane defined by an x-axis and a y-axis, used for plotting points. * **Intersection Point:** The point where two or more lines cross each other. This point represents the solution in graphical methods. ### Examples #### Example 1: Solve the following simultaneous equations graphically: 1. \( x + y = 5 \) 2. \( x - y = 1 \) * **Step 1: Rewrite in \( y = mx + c \) form.** * Equation 1: \( y = -x + 5 \) * Equation 2: \( y = x - 1 \) * **Step 2: Create a table of values.** * For \( y = -x + 5 \): | x | y \( = -x + 5 \) | (x, y) | | --- | ------------- | ------ | | -1 | 6 | (-1, 6) | | 0 | 5 | (0, 5) | | 2 | 3 | (2, 3) | * For \( y = x - 1 \): | x | y \( = x - 1 \) | (x, y) | | --- | ------------- | ------ | | -1 | -2 | (-1, -2) | | 0 | -1 | (0, -1) | | 2 | 1 | (2, 1) | * **Step 3 & 4: Plot points and draw lines.** (Imagine plotting these points on a graph paper and drawing the lines.) * **Step 5: Identify the point of intersection.** Looking at the graph, the lines intersect at the point \( (3, 2) \). So, \( x=3 \) and \( y=2 \). * **Step 6: Verify the solution.** * For Equation 1: \( x + y = 5 \) \( 3 + 2 = 5 \) (True) * For Equation 2: \( x - y = 1 \) \( 3 - 2 = 1 \) (True) Both equations are satisfied, so the solution \( (3, 2) \) is correct. #### Example 2: Solve graphically: 1. \( 2x + y = 4 \) 2. \( x - 2y = -3 \) * **Step 1: Rewrite in \( y = mx + c \) form.** * Equation 1: \( y = -2x + 4 \) * Equation 2: \( -2y = -x - 3 \Rightarrow 2y = x + 3 \Rightarrow y = \frac{1}{2}x + \frac{3}{2} \) * **Step 2: Create a table of values.** * For \( y = -2x + 4 \): | x | y \( = -2x + 4 \) | (x, y) | | --- | ------------- | ------ | | 0 | 4 | (0, 4) | | 1 | 2 | (1, 2) | | 2 | 0 | (2, 0) | * For \( y = \frac{1}{2}x + \frac{3}{2} \): (Choose x-values that make y an integer or easy to plot, e.g., even numbers for x to avoid fractions for y initially, or choose points around where you expect intersection based on the first table.) | x | y \( = \frac{1}{2}x + \frac{3}{2} \) | (x, y) | | --- | --------------------------- | ------ | | -3 | 0 | (-3, 0) | | 1 | 2 | (1, 2) | | 3 | 3 | (3, 3) | * **Step 3 & 4: Plot points and draw lines.** (Again, visualize plotting and drawing lines.) * **Step 5: Identify the point of intersection.** The lines intersect at \( (1, 2) \). So, \( x=1 \) and \( y=2 \). * **Step 6: Verify the solution.** * For Equation 1: \( 2x + y = 4 \) \( 2(1) + 2 = 2 + 2 = 4 \) (True) * For Equation 2: \( x - 2y = -3 \) \( 1 - 2(2) = 1 - 4 = -3 \) (True) The solution \( (1, 2) \) is correct. ### Exercises Solve the following simultaneous equations graphically. Remember to use graph paper, a ruler, and a sharp pencil. 1. \( y = x + 1 \) \( y = -x + 3 \) 2. \( x + 2y = 8 \) \( 3x - y = 3 \) 3. \( y = 2x - 3 \) \( x + y = 6 \) ### Assignment Continue practicing your graphical skills with these problems: 1. A store sells two types of shirts, A and B. Shirt A costs \( \$20 \) and Shirt B costs \( \$30 \). On a particular day, the store sold 10 shirts in total and collected \( \$240 \). Let \( x \) be the number of Shirt A sold and \( y \) be the number of Shirt B sold. * Write down two linear equations representing this situation. * Graph these equations and find the number of each type of shirt sold. 2. Solve the following system of equations graphically: \( 2x - y = 1 \) \( x + 3y = 10 \) 3. Consider the lines \( y = 3x - 2 \) and \( y = -x + 6 \). * Find the coordinates of the point of intersection graphically. * What would happen if the two lines were parallel? Could you find a solution? Explain. ### Summary In this lesson, you've learned how to solve linear simultaneous equations using the graphical method. This involves: * Rewriting equations into \( y = mx + c \) form. * Creating a table of values for each equation. * Plotting these points and drawing straight lines. * Identifying the point where the lines intersect, which gives the \( (x, y) \) solution. * Always remember to verify your solution by substituting the coordinates back into the original equations. This method provides a visual understanding of what it means for a pair of values to satisfy two equations simultaneously – it's the single point that lies on both lines. ## Graphical solutions to linear inequalities (Half plane) ### Introduction Welcome to this lesson on graphically solving linear inequalities! In previous grades, you learned how to solve linear equations, which have specific point solutions. Linear inequalities, however, represent a range of solutions. When we graph them, these solutions form a region called a 'half-plane' on the Cartesian coordinate system. This lesson will teach you how to visually represent these solutions, starting with single inequalities and moving on to systems of inequalities. ### Key Concepts #### What is a Linear Inequality? A linear inequality is a mathematical statement that compares two linear expressions using an inequality symbol. Unlike equations that use an 'equals' sign \( (=) \), inequalities use symbols such as: * \( < \) (less than) * \( > \) (greater than) * \( \leq \) (less than or equal to) * \( \geq \) (greater than or equal to) For example, \( y < 2x + 1 \) and \( 3x - y \geq 5 \) are linear inequalities. #### Graphing Linear Equations (Boundary Lines) To graph a linear inequality, the first step is to graph its related linear equation. This equation represents the *boundary line* of the solution region. For example, for the inequality \( y < 2x + 1 \), you would first graph the line \( y = 2x + 1 \). You can graph a linear equation by: 1. Finding the x- and y-intercepts. 2. Using the slope-intercept form \( y = mx + c \), where \( m \) is the gradient and \( c \) is the y-intercept. #### Solid vs. Dashed Lines The type of line used for the boundary is crucial for indicating whether the points on the line itself are part of the solution: * **Solid Line:** Used when the inequality includes "or equal to" (\( \leq \) or \( \geq \)). This means that all points on the boundary line *are* solutions to the inequality. * **Dashed (or Broken) Line:** Used when the inequality does *not* include "or equal to" (\( < \) or \( > \)). This means that points on the boundary line are *not* solutions; they only serve as a border for the solution region. #### Shading the Region (Half-Plane) A line divides the Cartesian plane into two regions, called half-planes. For a linear inequality, only one of these half-planes contains the solutions. We indicate the solution region by shading it. #### Test Point Method To determine which half-plane to shade, we use the test point method: 1. **Choose a Test Point:** Select any point that is *not* on the boundary line. The origin \( (0,0) \) is often the easiest choice if it does not lie on the line. 2. **Substitute Coordinates:** Substitute the x and y values of your test point into the original inequality. 3. **Evaluate:** * If the inequality holds true (e.g., \( 0 < 5 \)), then the region containing your test point is the solution. Shade that region. * If the inequality is false (e.g., \( 0 > 5 \)), then the region *opposite* to your test point is the solution. Shade the opposite region. #### Solving Systems of Linear Inequalities Graphically When you have two or more linear inequalities, it's called a system of linear inequalities. To find the solution to a system, you need to find the region that satisfies *all* inequalities simultaneously. 1. **Graph Each Inequality:** Graph each inequality separately on the same coordinate plane, carefully using solid/dashed lines and shading its solution region. 2. **Identify Overlap:** The solution to the system is the region where all the individual shaded areas overlap. This overlapping region is often called the **feasible region**. ### Vocabulary * **Linear Inequality:** A mathematical statement comparing two linear expressions using \( < \), \( > \), \( \leq \), or \( \geq \). * **Boundary Line:** The graph of the related linear equation, which separates the coordinate plane into two half-planes. * **Half-Plane:** One of the two regions into which a line divides a plane. The solution to a linear inequality lies in one of these half-planes. * **Test Point:** A point chosen to determine which side of the boundary line satisfies the inequality. * **Feasible Region:** The region on a graph that represents all common solutions to a system of linear inequalities. ### Examples #### Example 1: Graph \( y > 2x - 3 \) 1. **Graph the boundary line:** Graph \( y = 2x - 3 \). * When \( x=0 \), \( y = -3 \). Point: \( (0,-3) \). * When \( y=0 \), \( 0 = 2x - 3 \implies 2x = 3 \implies x = 1.5 \). Point: \( (1.5, 0) \). * Since the inequality is \( > \) (greater than), use a **dashed line**. 2. **Choose a test point:** Let's use \( (0,0) \). 3. **Substitute into inequality:** \( 0 > 2(0) - 3 \implies 0 > -3 \). This statement is TRUE. 4. **Shade:** Since the statement is true, shade the region that contains the test point \( (0,0) \) (the region above the dashed line). #### Example 2: Graph \( 3x + 2y \leq 6 \) 1. **Graph the boundary line:** Graph \( 3x + 2y = 6 \). * When \( x=0 \), \( 2y = 6 \implies y = 3 \). Point: \( (0,3) \). * When \( y=0 \), \( 3x = 6 \implies x = 2 \). Point: \( (2,0) \). * Since the inequality is \( \leq \) (less than or equal to), use a **solid line**. 2. **Choose a test point:** Let's use \( (0,0) \). 3. **Substitute into inequality:** \( 3(0) + 2(0) \leq 6 \implies 0 \leq 6 \). This statement is TRUE. 4. **Shade:** Since the statement is true, shade the region that contains the test point \( (0,0) \) (the region below the solid line). #### Example 3: Graph the solution for the system: \[ y \geq x - 1 \] \[ y < -2x + 4 \] 1. **Graph the first inequality: \( y \geq x - 1 \)** * Boundary line: \( y = x - 1 \) (solid line). * Test point \( (0,0) \): \( 0 \geq 0 - 1 \implies 0 \geq -1 \) (True). * Shade the region containing \( (0,0) \) (above the line). 2. **Graph the second inequality: \( y < -2x + 4 \)** * Boundary line: \( y = -2x + 4 \) (dashed line). * Test point \( (0,0) \): \( 0 < -2(0) + 4 \implies 0 < 4 \) (True). * Shade the region containing \( (0,0) \) (below the line). 3. **Identify the feasible region:** The solution to the system is the region where the shading from both inequalities overlaps. This will be the area above the solid line \( y = x - 1 \) and below the dashed line \( y = -2x + 4 \). ### Exercises Graph the solution region for each of the following inequalities or systems of inequalities. 1. Graph \( y \leq -x + 2 \) 2. Graph \( x - 2y > 4 \) 3. Graph the solution to the system: \[ y > x + 3 \] \[ y \leq -x + 1 \] 4. Graph the solution to the system, which forms a triangular region: \[ x \geq 0 \] \[ y \geq 0 \] \[ x + y \leq 5 \] ### Assignment Complete the following problems on a separate sheet of paper, showing all your working clearly. 1. Graph the inequality \( 4x + 3y < 12 \). 2. Graph the solution to the following system of inequalities: \[ y \geq \frac{1}{2}x - 2 \] \[ y \leq -3x + 5 \] \[ x \geq -1 \] 3. Determine whether the point \( (2, -1) \) is a solution to the inequality \( 3x - y < 8 \). Show your substitution and conclusion. ### Summary In this lesson, we explored how to represent linear inequalities graphically. Remember these key points: * A linear inequality represents a **half-plane** on the coordinate grid. * The **boundary line** is the graph of the related linear equation. * Use a **solid line** for inequalities with \( \leq \) or \( \geq \) (points on the line are included). * Use a **dashed line** for inequalities with \( < \) or \( > \) (points on the line are not included). * The **test point method** helps you determine which half-plane to shade. * For a **system of inequalities**, the solution is the **feasible region** where all individual solution regions overlap. ## Graphical solutions to simultaneous linear inequalities ### Introduction Welcome to this lesson on graphical solutions to simultaneous linear inequalities! In previous lessons, you learned how to solve single linear inequalities and how to represent them on a number line or a Cartesian plane. Today, we will extend this knowledge to solve problems involving two or more linear inequalities at the same time. This method is incredibly useful in various real-world scenarios, especially in optimization problems. ### Key Concepts #### What is a Linear Inequality? A linear inequality is a mathematical statement that compares two expressions using an inequality symbol. Unlike equations that use an equals sign \( = \), inequalities use symbols such as: * \( < \) (less than) * \( > \) (greater than) * \( \leq \) (less than or equal to) * \( \geq \) (greater than or equal to) Examples include \( y < 2x + 3 \) or \( x - y \geq 5 \). #### Graphing a Single Linear Inequality To graph a single linear inequality on a Cartesian plane, follow these steps: 1. **Change the inequality to an equation:** Replace the inequality sign with an equals sign (e.g., \( y < 2x + 3 \) becomes \( y = 2x + 3 \)). 2. **Draw the boundary line:** * If the inequality is \( < \) or \( > \), draw a **dashed (broken) line** to indicate that points on the line are *not* included in the solution. * If the inequality is \( \leq \) or \( \geq \), draw a **solid line** to indicate that points on the line *are* included in the solution. 3. **Choose a test point:** Select a point *not* on the line (e.g., \( (0,0) \) if it doesn't lie on the line). Substitute its coordinates into the original inequality. 4. **Shade the correct region:** * If the test point satisfies the inequality (makes it true), shade the region *containing* the test point. * If the test point does not satisfy the inequality (makes it false), shade the region *not containing* the test point. #### What are Simultaneous Linear Inequalities? Simultaneous linear inequalities are a set of two or more linear inequalities that must all be satisfied at the same time. The solution to a system of simultaneous linear inequalities is the region on the Cartesian plane where all individual inequalities overlap. #### Graphing Simultaneous Linear Inequalities To find the graphical solution to simultaneous linear inequalities, we combine the steps for graphing a single inequality: 1. **Graph each inequality separately:** For each inequality in the system, follow the steps outlined above to draw its boundary line and identify its solution region. 2. **Identify the common region:** The solution to the system of simultaneous linear inequalities is the region where the shaded areas of *all* individual inequalities overlap. This is often called the **feasible region**. 3. **Label the feasible region:** Clearly label or highlight the region that satisfies all inequalities. All points \( (x,y) \) within this feasible region (including on solid boundary lines) are solutions to the system. **Important Note:** When solving graphically, it is common practice to shade the *unwanted* region. This way, the desired feasible region (the solution) remains unshaded and is clearly visible. ### Vocabulary * **Linear Inequality:** A mathematical statement comparing two expressions using an inequality symbol \( (<, >, \leq, \geq) \). * **Boundary Line:** The line formed when an inequality is temporarily treated as an equation. It separates the Cartesian plane into two regions. * **Dashed Line (Broken Line):** A boundary line used for strict inequalities (\( < \) or \( > \)), indicating points on the line are not part of the solution. * **Solid Line:** A boundary line used for non-strict inequalities (\( \leq \) or \( \geq \)), indicating points on the line are part of the solution. * **Test Point:** A point chosen to determine which side of the boundary line satisfies the inequality. * **Shaded Region:** The area on the graph that represents all points satisfying an inequality. * **Simultaneous Linear Inequalities:** A set of two or more linear inequalities that must be satisfied concurrently. * **Feasible Region:** The region on the graph that satisfies all simultaneous linear inequalities. It's the intersection of all individual solution regions. ### Examples #### Example 1: Two Inequalities Find the region that satisfies the following inequalities: 1. \( y \geq x - 1 \) 2. \( y < -x + 3 \) **Solution Steps:** **Step 1: Graph \( y \geq x - 1 \)** * Boundary line: \( y = x - 1 \). This is a solid line because of \( \geq \). * Points: If \( x=0, y=-1 \) (0,-1); If \( y=0, x=1 \) (1,0). * Test point \( (0,0) \): \( 0 \geq 0 - 1 \Rightarrow 0 \geq -1 \) (True). So, shade the region *below* the line (containing \( (0,0) \)) if shading the unwanted region, or shade *above* if shading the wanted region. For this example, let's shade the unwanted region. **Step 2: Graph \( y < -x + 3 \)** * Boundary line: \( y = -x + 3 \). This is a dashed line because of \( < \). * Points: If \( x=0, y=3 \) (0,3); If \( y=0, x=3 \) (3,0). * Test point \( (0,0) \): \( 0 < -0 + 3 \Rightarrow 0 < 3 \) (True). So, shade the region *above* the line (not containing \( (0,0) \)) if shading the unwanted region. **Step 3: Identify the Feasible Region** The unshaded region on your graph, where the solutions of both inequalities overlap, is the feasible region. This region will be a triangle bounded by the two lines. #### Example 2: Three Inequalities Represent graphically the region that satisfies the following system of inequalities: 1. \( x \geq 0 \) 2. \( y \geq 0 \) 3. \( x + y \leq 4 \) **Solution Steps:** **Step 1: Graph \( x \geq 0 \)** * Boundary line: \( x = 0 \) (the y-axis). Solid line. * Shade the region to the left of the y-axis (unwanted region). **Step 2: Graph \( y \geq 0 \)** * Boundary line: \( y = 0 \) (the x-axis). Solid line. * Shade the region below the x-axis (unwanted region). **Step 3: Graph \( x + y \leq 4 \)** * Boundary line: \( x + y = 4 \). Solid line. * Points: If \( x=0, y=4 \) (0,4); If \( y=0, x=4 \) (4,0). * Test point \( (0,0) \): \( 0 + 0 \leq 4 \Rightarrow 0 \leq 4 \) (True). Shade the region *above* the line (unwanted region). **Step 4: Identify the Feasible Region** The feasible region will be the unshaded triangular area in the first quadrant, bounded by the x-axis, the y-axis, and the line \( x+y=4 \). This is a common region for linear programming problems. ### Exercises 1. On separate Cartesian planes, graph the solution region for each of the following single inequalities: a. \( y > 2x - 4 \) b. \( x + y \leq 5 \) c. \( y \geq -1 \) 2. Represent graphically the region that satisfies each system of simultaneous linear inequalities: a. \( y \geq x + 1 \) \( y < -2x + 5 \) b. \( x + y \leq 6 \) \( y \geq 2x \) \( x \geq 0 \) c. \( y \geq 0 \) \( x \leq 4 \) \( x + y \geq 2 \) ### Assignment Solve the following problems by drawing appropriate graphs on graph paper. 1. Find the region \( R \) defined by the inequalities: \( y \leq 3x + 1 \) \( y > -x + 5 \) \( x \geq 0 \) 2. A factory produces two types of toys, A and B. The production constraints are: * The number of toy A (\( x \)) must be at least 2. * The number of toy B (\( y \)) must be at least 3. * The total number of toys produced (\( x + y \)) cannot exceed 10. Write down these constraints as a system of linear inequalities and represent the feasible region for production graphically. 3. Identify the inequalities that define the unshaded region in the following graph (assume lines pass through integer coordinates shown on grid, not provided in this text-only response, but students would work from a visual graph). ### Summary In this lesson, we explored how to find graphical solutions to simultaneous linear inequalities. We learned that: * Each inequality has a boundary line (solid for \( \leq \) or \( \geq \), dashed for \( < \) or \( > \)). * A test point helps determine the region that satisfies an individual inequality. * The solution to simultaneous linear inequalities is the **feasible region**, which is the area where all individual inequalities' solution regions overlap. This is often represented as the unshaded region when shading the unwanted parts. This skill is fundamental for understanding and solving optimization problems in mathematics and real-world applications. ## Study of graphs of quadratic functions ### 1. Introduction Welcome to this lesson on quadratic functions! You've likely encountered linear functions before, whose graphs are straight lines. Quadratic functions are different; their graphs form beautiful curves called parabolas. Understanding these graphs is crucial in mathematics and various real-world applications, from physics (projectile motion) to engineering and economics. In this lesson, we will explore the key features of these graphs and learn how to sketch them accurately. ### 2. Key Concepts #### 2.1 What is a Quadratic Function? A quadratic function is a polynomial function of degree 2. It can be written in the general form: \[ y = ax^2 + bx + c \] where \( a, b, \) and \( c \) are real numbers, and \( a \neq 0 \). #### 2.2 The Parabola The graph of a quadratic function is a U-shaped curve called a parabola. ##### 2.2.1 Effect of the coefficient 'a' The value of \( a \) determines the direction and width of the parabola: * If \( a > 0 \) (positive), the parabola opens upwards. It has a **minimum point**. * If \( a < 0 \) (negative), the parabola opens downwards. It has a **maximum point**. * The larger the absolute value of \( a \) (\( |a| \)), the narrower the parabola. * The smaller the absolute value of \( a \), the wider the parabola. #### 2.3 Key Features of a Parabola Every parabola has several important features: ##### 2.3.1 Axis of Symmetry This is a vertical line that divides the parabola into two symmetrical halves. The equation of the axis of symmetry is given by: \[ x = -\frac{b}{2a} \] ##### 2.3.2 Vertex The vertex is the turning point of the parabola. It is either the lowest point (minimum) if \( a > 0 \) or the highest point (maximum) if \( a < 0 \). The coordinates of the vertex are found by substituting the x-value of the axis of symmetry into the function: \[ V = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \] ##### 2.3.3 y-intercept This is the point where the graph crosses the y-axis. It occurs when \( x = 0 \). For \( y = ax^2 + bx + c \), when \( x = 0 \), \( y = a(0)^2 + b(0) + c \), so \( y = c \). The y-intercept is \( (0, c) \). ##### 2.3.4 x-intercepts (Roots/Zeros) These are the points where the graph crosses the x-axis. They occur when \( y = 0 \). To find the x-intercepts, we solve the quadratic equation \( ax^2 + bx + c = 0 \). There are three possibilities based on the **discriminant** (\( \Delta \)): \[ \Delta = b^2 - 4ac \] * If \( \Delta > 0 \): The graph has **two distinct real x-intercepts**. It crosses the x-axis at two different points. The roots can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] * If \( \Delta = 0 \): The graph has **one repeated real x-intercept**. It touches the x-axis at exactly one point (the vertex lies on the x-axis). The root is \( x = -\frac{b}{2a} \). * If \( \Delta < 0 \): The graph has **no real x-intercepts**. It does not cross or touch the x-axis. #### 2.4 Steps for Sketching a Quadratic Graph To sketch the graph of a quadratic function \( y = ax^2 + bx + c \), follow these steps: 1. **Determine the direction:** Check the sign of \( a \). If \( a > 0 \), it opens upwards. If \( a < 0 \), it opens downwards. 2. **Find the y-intercept:** Set \( x = 0 \) to find \( (0, c) \). 3. **Find the axis of symmetry:** Use the formula \( x = -\frac{b}{2a} \). 4. **Find the vertex:** Substitute the x-value of the axis of symmetry into the function to find the y-coordinate. This gives you \( \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \). 5. **Find the x-intercepts (if any):** Set \( y = 0 \) and solve \( ax^2 + bx + c = 0 \) using factorization or the quadratic formula. Check the discriminant first. 6. **Plot points and sketch:** Plot the y-intercept, vertex, and x-intercepts. If needed, find a few additional points by choosing x-values on either side of the axis of symmetry (e.g., \( x=1 \) and \( x=-1 \) if the axis of symmetry is \( x=0 \)). Draw a smooth, symmetrical curve through these points. ### 3. Vocabulary * **Quadratic Function:** A polynomial function of degree 2, typically \( y = ax^2 + bx + c \). * **Parabola:** The U-shaped graph of a quadratic function. * **Vertex:** The turning point of a parabola; it's either the maximum or minimum point. * **Axis of Symmetry:** A vertical line that divides the parabola into two congruent halves. * **x-intercepts (Roots/Zeros):** The points where the graph crosses or touches the x-axis (where \( y = 0 \)). * **y-intercept:** The point where the graph crosses the y-axis (where \( x = 0 \)). * **Discriminant (\( \Delta \)):** The expression \( b^2 - 4ac \) used to determine the nature of the roots of a quadratic equation. * **Minimum Point:** The lowest point on a parabola that opens upwards. * **Maximum Point:** The highest point on a parabola that opens downwards. ### 4. Examples #### Example 1: Sketch the graph of \( y = x^2 - 6x + 5 \). 1. **Direction:** \( a = 1 > 0 \), so the parabola opens upwards. 2. **y-intercept:** Set \( x = 0 \) \( \Rightarrow y = 0^2 - 6(0) + 5 = 5 \). The y-intercept is \( (0, 5) \). 3. **Axis of symmetry:** \( x = -\frac{b}{2a} = -\frac{-6}{2(1)} = \frac{6}{2} = 3 \). The axis of symmetry is \( x = 3 \). 4. **Vertex:** Substitute \( x = 3 \) into the function: \( y = (3)^2 - 6(3) + 5 = 9 - 18 + 5 = -4 \). The vertex is \( (3, -4) \). 5. **x-intercepts:** Set \( y = 0 \) \( \Rightarrow x^2 - 6x + 5 = 0 \). Factor: \( (x-1)(x-5) = 0 \) \( \Rightarrow x = 1 \) or \( x = 5 \). The x-intercepts are \( (1, 0) \) and \( (5, 0) \). 6. **Sketch:** Plot the points \( (0, 5) \), \( (3, -4) \), \( (1, 0) \), \( (5, 0) \) and draw a smooth parabola opening upwards. #### Example 2: Sketch the graph of \( y = -2x^2 - 4x + 6 \). 1. **Direction:** \( a = -2 < 0 \), so the parabola opens downwards. 2. **y-intercept:** Set \( x = 0 \) \( \Rightarrow y = -2(0)^2 - 4(0) + 6 = 6 \). The y-intercept is \( (0, 6) \). 3. **Axis of symmetry:** \( x = -\frac{b}{2a} = -\frac{-4}{2(-2)} = -\frac{-4}{-4} = -1 \). The axis of symmetry is \( x = -1 \). 4. **Vertex:** Substitute \( x = -1 \) into the function: \( y = -2(-1)^2 - 4(-1) + 6 = -2(1) + 4 + 6 = -2 + 4 + 6 = 8 \). The vertex is \( (-1, 8) \). 5. **x-intercepts:** Set \( y = 0 \) \( \Rightarrow -2x^2 - 4x + 6 = 0 \). Divide by -2: \( x^2 + 2x - 3 = 0 \). Factor: \( (x+3)(x-1) = 0 \) \( \Rightarrow x = -3 \) or \( x = 1 \). The x-intercepts are \( (-3, 0) \) and \( (1, 0) \). 6. **Sketch:** Plot the points \( (0, 6) \), \( (-1, 8) \), \( (-3, 0) \), \( (1, 0) \) and draw a smooth parabola opening downwards. #### Example 3: Determine the nature of the roots for \( y = 3x^2 - 5x + 4 \). Here, \( a = 3, b = -5, c = 4 \). Calculate the discriminant: \( \Delta = b^2 - 4ac = (-5)^2 - 4(3)(4) = 25 - 48 = -23 \). Since \( \Delta = -23 < 0 \), there are no real x-intercepts (the graph does not cross the x-axis). ### 5. Exercises 1. For each quadratic function, state whether its graph opens upwards or downwards: a) \( y = 2x^2 - 3x + 1 \) b) \( y = -x^2 + 5x - 7 \) c) \( y = \frac{1}{2}x^2 + 4 \) 2. Find the coordinates of the y-intercept for each function: a) \( y = x^2 + 7x - 10 \) b) \( y = -3x^2 + 2x + 5 \) 3. Find the equation of the axis of symmetry and the coordinates of the vertex for: a) \( y = x^2 - 8x + 12 \) b) \( y = -2x^2 - 12x - 16 \) 4. Determine the number of x-intercepts for each function using the discriminant: a) \( y = x^2 + 2x - 3 \) b) \( y = 4x^2 - 12x + 9 \) c) \( y = -x^2 + x - 1 \) 5. Sketch the graph of \( y = x^2 - 2x - 8 \), clearly showing the vertex, y-intercept, and x-intercepts. ### 6. Assignment 1. A ball is thrown upwards, and its height \( h \) in meters after \( t \) seconds is given by the function \( h(t) = -5t^2 + 20t + 25 \). a) What is the initial height of the ball? b) Find the maximum height reached by the ball. c) How long does it take for the ball to reach its maximum height? d) How long does it take for the ball to hit the ground? 2. The profit \( P \) (in thousands of dollars) for a company selling \( x \) units of a certain product is given by \( P(x) = -x^2 + 10x - 16 \). a) What is the profit if no units are sold? b) How many units must be sold to maximize profit? c) What is the maximum profit? d) For what number of units sold does the company break even (profit is zero)? 3. Sketch the graph of \( y = 2x^2 + 4x + 3 \). Make sure to indicate the vertex and y-intercept. Explain why this graph has no x-intercepts. ### 7. Summary In this lesson, we have explored the graphs of quadratic functions, which are known as parabolas. We learned that the coefficient \( a \) determines whether the parabola opens upwards (minimum point) or downwards (maximum point). Key features like the y-intercept \( (0, c) \), the axis of symmetry \( x = -\frac{b}{2a} \), and the vertex \( \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \) are essential for sketching. We also understood how to find x-intercepts by solving \( ax^2 + bx + c = 0 \) and how the discriminant \( \Delta = b^2 - 4ac \) tells us the number of real roots (zero, one, or two). Mastering these concepts allows us to analyze and interpret quadratic functions effectively, connecting algebraic expressions to their visual representations. ## The discriminant and the nature of roots of a quadratic equation ### Introduction Welcome to this lesson on quadratic equations! In previous lessons, you learned how to solve quadratic equations using various methods like factorization, completing the square, and the quadratic formula. Today, we will dive deeper into understanding the *nature* of the solutions (or roots) of a quadratic equation *without* actually solving it completely. This is possible by using a special part of the quadratic formula called the **discriminant**. ### Key Concepts #### What is a Quadratic Equation? A quadratic equation is an equation of the second degree, meaning it contains at least one term in which the unknown variable is squared. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are real numbers, and \( a \neq 0 \). #### The Quadratic Formula When a quadratic equation cannot be easily factored, we use the quadratic formula to find its roots (solutions). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] #### The Discriminant The **discriminant** is the expression under the square root sign in the quadratic formula. It is denoted by the Greek letter delta, \( \Delta \) (or sometimes simply \( D \)). \[ \Delta = b^2 - 4ac \] The value of the discriminant tells us a lot about the nature of the roots of the quadratic equation without having to calculate the roots themselves. #### Nature of Roots based on the Discriminant The discriminant \( \Delta = b^2 - 4ac \) helps us determine whether the roots are real or complex, and if real, whether they are distinct or equal. There are three main cases: 1. **If \( \Delta > 0 \) (Discriminant is positive):** * The equation has **two distinct real roots**. * This means the graph of the quadratic function \( y = ax^2 + bx + c \) intersects the x-axis at two different points. * **Sub-case:** If \( \Delta \) is also a perfect square (e.g., 4, 9, 16, 25, ...), the two distinct real roots are **rational**. * **Sub-case:** If \( \Delta \) is not a perfect square, the two distinct real roots are **irrational**. 2. **If \( \Delta = 0 \) (Discriminant is zero):** * The equation has **two equal real roots** (also called one repeated real root). * This means the graph of the quadratic function touches the x-axis at exactly one point (it is tangent to the x-axis). * These roots are always **rational**. 3. **If \( \Delta < 0 \) (Discriminant is negative):** * The equation has **no real roots**. * Instead, it has two distinct **complex (or imaginary) roots**. * This means the graph of the quadratic function does not intersect the x-axis at all. #### Summary Table | Value of Discriminant (\( \Delta = b^2 - 4ac \)) | Nature of Roots | | :------------------------------------------ | :-------------------------------------------------- | | \( \Delta > 0 \) and perfect square | Two distinct, real, and rational roots | | \( \Delta > 0 \) and not a perfect square | Two distinct, real, and irrational roots | | \( \Delta = 0 \) | Two equal, real, and rational roots | | \( \Delta < 0 \) | No real roots (Two distinct complex/imaginary roots) | ### Vocabulary * **Quadratic Equation:** An equation of the form \( ax^2 + bx + c = 0 \) where \( a \neq 0 \). * **Roots (or Solutions):** The values of the variable that satisfy the quadratic equation. * **Discriminant (\( \Delta \)):** The expression \( b^2 - 4ac \) from the quadratic formula, used to determine the nature of the roots. * **Real Roots:** Roots that are real numbers (can be rational or irrational). * **Complex/Imaginary Roots:** Roots that involve the imaginary unit \( i = \sqrt{-1} \). They are of the form \( p \pm qi \). * **Distinct Roots:** Roots that have different values. * **Equal Roots (or Repeated Roots):** Roots that have the same value. * **Rational Roots:** Real roots that can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \). * **Irrational Roots:** Real roots that cannot be expressed as a simple fraction (e.g., \( \sqrt{2} \), \( 1 + \sqrt{3} \)). * **Perfect Square:** A number that is the square of an integer (e.g., 1, 4, 9, 16, 25). ### Examples Let's determine the nature of the roots for the following quadratic equations: #### Example 1: Two Distinct Real and Rational Roots Equation: \( x^2 - 5x + 6 = 0 \) 1. Identify \( a, b, c \): \( a = 1 \), \( b = -5 \), \( c = 6 \) 2. Calculate the discriminant: \( \Delta = b^2 - 4ac = (-5)^2 - 4(1)(6) \) \( \Delta = 25 - 24 \) \( \Delta = 1 \) 3. Interpret the result: Since \( \Delta = 1 \) and \( 1 > 0 \), there are two distinct real roots. Also, since \( 1 \) is a perfect square (\( 1^2 = 1 \)), the roots are rational. #### Example 2: Two Distinct Real and Irrational Roots Equation: \( 2x^2 + 4x - 5 = 0 \) 1. Identify \( a, b, c \): \( a = 2 \), \( b = 4 \), \( c = -5 \) 2. Calculate the discriminant: \( \Delta = b^2 - 4ac = (4)^2 - 4(2)(-5) \) \( \Delta = 16 - (-40) \) \( \Delta = 16 + 40 \) \( \Delta = 56 \) 3. Interpret the result: Since \( \Delta = 56 \) and \( 56 > 0 \), there are two distinct real roots. Since \( 56 \) is not a perfect square, the roots are irrational. #### Example 3: Two Equal Real and Rational Roots Equation: \( x^2 - 6x + 9 = 0 \) 1. Identify \( a, b, c \): \( a = 1 \), \( b = -6 \), \( c = 9 \) 2. Calculate the discriminant: \( \Delta = b^2 - 4ac = (-6)^2 - 4(1)(9) \) \( \Delta = 36 - 36 \) \( \Delta = 0 \) 3. Interpret the result: Since \( \Delta = 0 \), there are two equal real roots. These roots are also rational. #### Example 4: No Real Roots (Two Distinct Complex Roots) Equation: \( 3x^2 - 2x + 1 = 0 \) 1. Identify \( a, b, c \): \( a = 3 \), \( b = -2 \), \( c = 1 \) 2. Calculate the discriminant: \( \Delta = b^2 - 4ac = (-2)^2 - 4(3)(1) \) \( \Delta = 4 - 12 \) \( \Delta = -8 \) 3. Interpret the result: Since \( \Delta = -8 \) and \( -8 < 0 \), there are no real roots. The equation has two distinct complex (imaginary) roots. ### Exercises Determine the nature of the roots for each of the following quadratic equations: 1. \( x^2 + 8x + 16 = 0 \) 2. \( x^2 - 3x - 10 = 0 \) 3. \( 3x^2 + x + 2 = 0 \) 4. \( 2x^2 - 7x + 3 = 0 \) 5. \( x^2 + 5x + 1 = 0 \) 6. \( 4x^2 - 12x + 9 = 0 \) ### Assignment 1. For what value(s) of \( k \) does the equation \( x^2 + kx + 4 = 0 \) have: a. Two distinct real roots? b. Two equal real roots? c. No real roots? 2. Find the value of \( p \) if the quadratic equation \( px^2 - 4x + 1 = 0 \) has two equal real roots. 3. Show that the equation \( x^2 - 2x + 5 = 0 \) has no real roots. 4. Determine the nature of the roots of the equation \( 5x^2 + 2x - 3 = 0 \) and then solve the equation to confirm your answer. ### Summary In this lesson, we learned about the discriminant, \( \Delta = b^2 - 4ac \), which is a powerful tool derived from the quadratic formula. By simply calculating the value of the discriminant, we can determine the nature of the roots of any quadratic equation \( ax^2 + bx + c = 0 \) without solving for \( x \): * If \( \Delta > 0 \): Two distinct real roots (rational if \( \Delta \) is a perfect square, irrational otherwise). * If \( \Delta = 0 \): Two equal real (and rational) roots. * If \( \Delta < 0 \): No real roots (two distinct complex roots). Understanding the discriminant helps us predict the type of solutions we expect and can be very useful in various mathematical and real-world problems. Keep practicing to master this concept! ## Graphical solutions to quadratic equations. ### Introduction Welcome to this lesson on graphical solutions to quadratic equations! You have previously learned how to solve quadratic equations using methods such as factorization and the quadratic formula. In this lesson, we will explore another powerful method: solving quadratic equations by graphing them. This method not only helps us find the solutions (also known as roots) but also gives us a visual understanding of the behavior of quadratic functions. ### Key Concepts #### What is a Quadratic Equation? A quadratic equation is an equation that can be written in the standard form: \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are real numbers, and \( a \neq 0 \). The highest power of the variable \( x \) is 2. #### The Graph of a Quadratic Equation: The Parabola When we plot a quadratic equation, we get a characteristic U-shaped curve called a **parabola**. The shape and orientation of the parabola depend on the coefficient \( a \). * If \( a > 0 \), the parabola opens upwards (like a smile). * If \( a < 0 \), the parabola opens downwards (like a frown). Key features of a parabola: * **Vertex:** The highest or lowest point on the parabola. It's the turning point. * **Axis of Symmetry:** A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. * **x-intercepts:** The points where the parabola crosses or touches the x-axis. **These are the solutions (roots) of the quadratic equation.** * **y-intercept:** The point where the parabola crosses the y-axis (when \( x = 0 \), so it is \( (0, c) \) from the equation \( y = ax^2 + bx + c \)). #### Steps to Solve a Quadratic Equation Graphically To solve a quadratic equation of the form \( ax^2 + bx + c = 0 \) graphically, follow these steps: 1. **Rewrite the Equation:** Convert the quadratic equation into its corresponding quadratic function form: \( y = ax^2 + bx + c \). 2. **Create a Table of Values:** Choose a range of \( x \)-values (e.g., from -3 to 3, or -4 to 4, depending on the equation) and calculate the corresponding \( y \)-values using the function \( y = ax^2 + bx + c \). It's helpful to include the vertex in your range if possible. 3. **Plot the Points:** Draw a Cartesian plane and plot the points \( (x, y) \) from your table. 4. **Draw the Parabola:** Connect the plotted points with a smooth, continuous curve to form the parabola. Extend the curve beyond your plotted points if it clearly follows the parabolic shape.