6. Equations

# 6. Equations ## Solutions to simple linear equations # Solutions to Simple Linear Equations\n\n## Introduction\nThis lesson will introduce you to simple linear equations, how to identify them, and various methods to find their solutions. Equations are fundamental tools in mathematics used to represent relationships and solve problems in many real-world situations.\n\n## Key Concepts\n\n### What is an Equation?\nAn equation is a mathematical statement that shows two expressions are equal. It always contains an **equals sign** \( (=) \).\n* **Example:** \( 2x + 3 = 7 \)\n * Left side: \( 2x + 3 \)\n * Right side: \( 7 \)\n\n### What is a Linear Equation?\nA linear equation is an equation where the highest power of the variable is 1. There are no squared terms \( (x^2) \), cubed terms \( (x^3) \), or variables multiplied by each other \( (xy) \).\n* **Example:** \( x + 5 = 12 \) (variable \( x \) has power 1)\n* **Non-Example:** \( x^2 - 4 = 0 \) (variable \( x \) has power 2, not linear)\n\n### What is a Solution to an Equation?\nThe solution to an equation is the value(s) of the variable that makes the equation true. When you substitute the solution back into the original equation, both sides of the equation will be equal.\n* **Example:** For \( x + 3 = 7 \), the solution is \( x = 4 \) because \( 4 + 3 = 7 \) is true.\n\n### Methods to Solve Simple Linear Equations\nThe main goal is to **isolate the variable** on one side of the equation. We do this by performing **inverse operations**.\n\n#### 1. Balancing Method (Using Addition and Subtraction)\nThe principle is: whatever you do to one side of the equation, you must do to the other side to keep it balanced.\n* **To remove a number added to the variable:** Subtract that number from both sides.\n* **To remove a number subtracted from the variable:** Add that number to both sides.\n\n**Example 1.1:** Solve \( x + 5 = 12 \)\n* To isolate \( x \), we need to get rid of \( +5 \). The inverse operation of addition is subtraction.\n* Subtract 5 from both sides:\n \[ x + 5 - 5 = 12 - 5 \]\n \[ x = 7 \]\n* **Check:** Substitute \( x = 7 \) back into the original equation: \( 7 + 5 = 12 \). This is true, so the solution is correct.\n\n**Example 1.2:** Solve \( y - 3 = 8 \)\n* To isolate \( y \), we need to get rid of \( -3 \). The inverse operation of subtraction is addition.\n* Add 3 to both sides:\n \[ y - 3 + 3 = 8 + 3 \]\n \[ y = 11 \]\n* **Check:** \( 11 - 3 = 8 \). True.\n\n#### 2. Balancing Method (Using Multiplication and Division)\n* **To remove a number multiplying the variable:** Divide both sides by that number.\n* **To remove a number dividing the variable:** Multiply both sides by that number.\n\n**Example 2.1:** Solve \( 3x = 15 \)\n* To isolate \( x \), we need to get rid of the \( 3 \) that is multiplying \( x \). The inverse operation of multiplication is division.\n* Divide both sides by 3:\n \[ \frac{3x}{3} = \frac{15}{3} \]\n \[ x = 5 \]\n* **Check:** \( 3(5) = 15 \). True.\n\n**Example 2.2:** Solve \( \frac{m}{4} = 6 \)\n* To isolate \( m \), we need to get rid of the \( 4 \) that is dividing \( m \). The inverse operation of division is multiplication.\n* Multiply both sides by 4:\n \[ 4 \times \frac{m}{4} = 6 \times 4 \]\n \[ m = 24 \]\n* **Check:** \( \frac{24}{4} = 6 \). True.\n\n#### 3. Solving Multi-Step Linear Equations\nSometimes you need to use more than one inverse operation. A good strategy is to:\n1. Deal with addition/subtraction first.\n2. Then deal with multiplication/division.\n\n**Example 3.1:** Solve \( 2x + 1 = 9 \)\n1. Subtract 1 from both sides (to remove \( +1 \)):\n \[ 2x + 1 - 1 = 9 - 1 \]\n \[ 2x = 8 \]\n2. Divide both sides by 2 (to remove multiplication by 2):\n \[ \frac{2x}{2} = \frac{8}{2} \]\n \[ x = 4 \]\n* **Check:** \( 2(4) + 1 = 8 + 1 = 9 \). True.\n\n**Example 3.2:** Solve \( \frac{x}{3} - 2 = 5 \)\n1. Add 2 to both sides (to remove \( -2 \)):\n \[ \frac{x}{3} - 2 + 2 = 5 + 2 \]\n \[ \frac{x}{3} = 7 \]\n2. Multiply both sides by 3 (to remove division by 3):\n \[ 3 \times \frac{x}{3} = 7 \times 3 \]\n \[ x = 21 \]\n* **Check:** \( \frac{21}{3} - 2 = 7 - 2 = 5 \). True.\n\n## Vocabulary\n* **Equation:** A mathematical statement showing that two expressions are equal, containing an equals sign \( (=) \).\n* **Variable:** A symbol (usually a letter like \( x \), \( y \), \( m \)) used to represent an unknown number.\n* **Constant:** A fixed value in an expression or equation that does not change (e.g., 5, -2, 12).\n* **Solution (of an equation):** The value(s) of the variable that makes the equation a true statement.\n* **Linear Equation:** An equation where the highest power of the variable is 1.\n* **Inverse Operation:** An operation that "undoes" another operation (e.g., addition is the inverse of subtraction, multiplication is the inverse of division).\n* **Isolate the variable:** To get the variable by itself on one side of the equation.\n\n## Examples\nHere are more examples demonstrating the steps:\n\n**Example 1:** Solve for \( p \) in \( p + 7 = 15 \)\n* Subtract 7 from both sides:\n \[ p + 7 - 7 = 15 - 7 \]\n \[ p = 8 \]\n\n**Example 2:** Solve for \( q \) in \( q - 10 = 2 \)\n* Add 10 to both sides:\n \[ q - 10 + 10 = 2 + 10 \]\n \[ q = 12 \]\n\n**Example 3:** Solve for \( r \) in \( 5r = 45 \)\n* Divide both sides by 5:\n \[ \frac{5r}{5} = \frac{45}{5} \]\n \[ r = 9 \]\n\n**Example 4:** Solve for \( s \) in \( \frac{s}{2} = 11 \)\n* Multiply both sides by 2:\n \[ 2 \times \frac{s}{2} = 11 \times 2 \]\n \[ s = 22 \]\n\n**Example 5:** Solve for \( t \) in \( 4t - 3 = 21 \)\n1. Add 3 to both sides:\n \[ 4t - 3 + 3 = 21 + 3 \]\n \[ 4t = 24 \]\n2. Divide both sides by 4:\n \[ \frac{4t}{4} = \frac{24}{4} \]\n \[ t = 6 \]\n\n## Exercises\nSolve the following simple linear equations. Show your steps.\n\n1. \( x + 8 = 17 \)\n2. \( y - 4 = 13 \)\n3. \( 6m = 30 \)\n4. \( \frac{n}{5} = 9 \)\n5. \( 2p + 3 = 11 \)\n6. \( 3q - 5 = 16 \)\n7. \( \frac{r}{2} + 1 = 7 \)\n8. \( 5s - 7 = 3 \)\n9. \( 10 + 4t = 30 \)\n10. \( \frac{v}{3} - 6 = 2 \)\n\n## Assignment\nSolve the following equations at home. Remember to check your answers!\n\n1. \( a + 12 = 25 \)\n2. \( b - 9 = 1 \)\n3. \( 7c = 42 \)\n4. \( \frac{d}{6} = 3 \)\n5. \( 4e + 5 = 29 \)\n6. \( 8f - 1 = 23 \)\n7. \( \frac{g}{4} + 3 = 5 \)\n8. \( 9h - 10 = 8 \)\n9. \( 15 = 3j - 6 \)\n10. \( \frac{k}{5} + 7 = 10 \)\n\n## Summary\nIn this lesson, we learned about simple linear equations and how to find their solutions.\n* An **equation** shows two expressions are equal using an \( (=) \) sign.\n* A **linear equation** has variables raised to the power of 1.\n* The **solution** is the value that makes the equation true.\n* We solve equations by using **inverse operations** to **isolate the variable**.\n * Add to undo subtraction.\n * Subtract to undo addition.\n * Multiply to undo division.\n * Divide to undo multiplication.\n* For multi-step equations, generally deal with addition/subtraction first, then multiplication/division. ## Inequalities; # Inequalities\n\n## Introduction\nWelcome to this lesson on Inequalities! In mathematics, we often deal with equations where two expressions are equal. However, there are many situations where quantities are not equal, but instead, one is greater than, less than, greater than or equal to, or less than or equal to the other. These relationships are called inequalities. Understanding inequalities is crucial for solving real-world problems involving limits, ranges, and comparisons.\n\n## Key Concepts\n\n### What is an Inequality?\nAn inequality is a mathematical statement that compares two expressions using an inequality symbol. Unlike equations that show equality (using '='), inequalities show a difference in value.\n\n### Inequality Symbols\nThere are four main inequality symbols you need to know:\n* \\( > \\) : **Greater than** (e.g., \\( 5 > 2 \\) means 5 is greater than 2)\n* \\( < \\) : **Less than** (e.g., \\( 3 < 7 \\) means 3 is less than 7)\n* \\( \\ge \\) : **Greater than or equal to** (e.g., \\( x \\ge 4 \\) means x can be 4 or any number larger than 4)\n* \\( \\le \\) : **Less than or equal to** (e.g., \\( y \\le 10 \\) means y can be 10 or any number smaller than 10)\n\n### Properties of Inequalities\nSolving inequalities is similar to solving equations, but there's a very important rule to remember, especially when multiplying or dividing.\n\n#### Addition and Subtraction Property\nIf you add or subtract the same number from both sides of an inequality, the direction of the inequality sign **does not change**.\n* If \\( a < b \\), then \\( a + c < b + c \\)\n* If \\( a < b \\), then \\( a - c < b - c \\)\n\n#### Multiplication and Division Property\n* **Multiplying or Dividing by a Positive Number:** If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality sign **does not change**.\n * If \\( a < b \\) and \\( c > 0 \\), then \\( ac < bc \\)\n * If \\( a < b \\) and \\( c > 0 \\), then \\( \\frac{a}{c} < \\frac{b}{c} \\)\n* **Multiplying or Dividing by a Negative Number:** If you multiply or divide both sides of an inequality by a negative number, you **MUST reverse the direction of the inequality sign**.\n * If \\( a < b \\) and \\( c < 0 \\), then \\( ac > bc \\)\n * If \\( a < b \\) and \\( c < 0 \\), then \\( \\frac{a}{c} > \\frac{b}{c} \\)\n This rule is crucial for solving inequalities correctly.\n\n### Solving Linear Inequalities\nTo solve an inequality, you want to isolate the variable on one side of the inequality sign, just like with equations. Remember the properties above!\n\n### Representing Inequalities on a Number Line\nA number line is a visual way to show the solution set of an inequality.\n* **Open Circle ( \\( \\circ \\) ):** Used for 'less than' (\\( < \\)) or 'greater than' (\\( > \\)) inequalities. It means the number itself is **not included** in the solution.\n* **Closed Circle ( \\( \\bullet \\) ):** Used for 'less than or equal to' (\\( \\le \\)) or 'greater than or equal to' (\\( \\ge \\)) inequalities. It means the number itself **is included** in the solution.\n* **Shading:** Draw an arrow or shade the part of the number line that represents all possible solutions.\n\n## Vocabulary\n* **Inequality:** A mathematical statement that shows two quantities are not equal, using symbols like \\( <, >, \\le, \\ge \\).\n* **Solution Set:** The set of all values that make an inequality true.\n* **Number Line:** A visual representation of numbers as points on a straight line.\n\n## Examples\nLet's work through some examples.\n\n### Example 1: Basic Addition/Subtraction\nSolve \\( x + 3 < 7 \\) and represent the solution on a number line.\n1. Subtract 3 from both sides: \\( x + 3 - 3 < 7 - 3 \\)\n2. Simplify: \\( x < 4 \\)\n\nOn a number line, you would draw an open circle at 4 and shade to the left.\n\n### Example 2: Basic Multiplication/Division (Positive)\nSolve \\( 2x \\ge 10 \\) and represent the solution on a number line.\n1. Divide both sides by 2 (a positive number, so the sign doesn't flip): \\( \\frac{2x}{2} \\ge \\frac{10}{2} \\)\n2. Simplify: \\( x \\ge 5 \\)\n\nOn a number line, you would draw a closed circle at 5 and shade to the right.\n\n### Example 3: Multiplication/Division (Negative)\nSolve \\( -3x + 4 > 10 \\) and represent the solution on a number line.\n1. Subtract 4 from both sides: \\( -3x + 4 - 4 > 10 - 4 \\)\n2. Simplify: \\( -3x > 6 \\)\n3. Divide both sides by -3. **REMEMBER TO FLIP THE INEQUALITY SIGN!** \\( \\frac{-3x}{-3} < \\frac{6}{-3} \\)\n4. Simplify: \\( x < -2 \\)\n\nOn a number line, you would draw an open circle at -2 and shade to the left.\n\n### Example 4: Multiple Steps\nSolve \\( 5x - 7 \\le 3x + 9 \\) and represent the solution on a number line.\n1. Subtract \\( 3x \\) from both sides: \\( 5x - 3x - 7 \\le 3x - 3x + 9 \\)\n2. Simplify: \\( 2x - 7 \\le 9 \\)\n3. Add 7 to both sides: \\( 2x - 7 + 7 \\le 9 + 7 \\)\n4. Simplify: \\( 2x \\le 16 \\)\n5. Divide both sides by 2: \\( \\frac{2x}{2} \\le \\frac{16}{2} \\)\n6. Simplify: \\( x \\le 8 \\)\n\nOn a number line, you would draw a closed circle at 8 and shade to the left.\n\n## Exercises\nSolve each inequality and represent the solution on a number line.\n1. \\( y - 5 > 2 \\)\n2. \\( 4m \\le 24 \\)\n3. \\( -x + 8 \\ge 3 \\)\n4. \\( 2p + 1 < 13 \\)\n5. \\( 7 - 3k \\le 1 \\)\n\n## Assignment\nSolve the following inequalities. Show your steps clearly and represent your final answer on a number line.\n1. \\( x + 12 \\ge 20 \\)\n2. \\( \\frac{a}{3} - 1 < 4 \\)\n3. \\( 15 - 2b > 9 \\)\n4. \\( 4(y - 3) \\le 8 \\)\n5. \\( \\frac{5x + 1}{2} > -7 \\)\n6. A taxi charges a base fare of $3 and an additional $0.50 per kilometre. If a customer has a maximum of $10 to spend, what is the maximum distance (in kilometres) they can travel?\n\n## Summary\nToday, we learned about inequalities, which are mathematical statements comparing two expressions that are not necessarily equal. We explored the four main inequality symbols (\\( <, >, \\le, \\ge \\)). Key properties include: adding or subtracting a number doesn't change the sign, multiplying or dividing by a positive number doesn't change the sign, but **multiplying or dividing by a negative number reverses the sign**. We also practised solving linear inequalities and representing their solution sets on a number line using open (not included) or closed (included) circles and shading. Mastering inequalities helps us describe and solve problems with ranges and limits. ## Solutions to Linear inequalities ### Introduction Welcome to this lesson on solutions to linear inequalities! In previous lessons, you learned how to solve linear equations, where the goal was to find a specific value for an unknown variable (e.g., \( x = 5 \)). Today, we will explore a related but different concept: linear inequalities. Instead of finding a single value, we will find a range of values that satisfy a given condition. This range is called the solution set, and we'll learn how to represent it both algebraically and graphically on a number line. ### Key Concepts #### What is an Inequality? An inequality is a mathematical statement that compares two expressions using an inequality sign. Unlike equations which use an equals sign (\( = \)), inequalities tell us if one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. Here are the common inequality signs: * \( < \): Less than * \( > \): Greater than * \( \le \): Less than or equal to * \( \ge \): Greater than or equal to #### Linear Inequalities A linear inequality is an inequality that involves a variable raised to the power of 1. It does not contain any variables multiplied together or variables with exponents other than 1. Examples of linear inequalities: * \( x + 3 > 7 \) * \( 2y - 5 \le 1 \) * \( 4 - z \ge 10 \) * \( \frac{a}{2} < 6 \) #### Solving Linear Inequalities Solving linear inequalities is very similar to solving linear equations. We use inverse operations to isolate the variable. The main operations are: * **Addition and Subtraction:** You can add or subtract the same number from both sides of an inequality without changing the direction of the inequality sign. * If \( x - 2 < 5 \), then \( x - 2 + 2 < 5 + 2 \), which means \( x < 7 \). * **Multiplication and Division 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 \( 3x > 9 \), then \( \frac{3x}{3} > \frac{9}{3} \), which means \( x > 3 \). * **Crucial Difference: Multiplication and Division by a Negative Number:** **When you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality sign.** This is the most important rule to remember when solving inequalities. * If \( -2x < 6 \), then dividing by \( -2 \) means we must reverse the sign: \[ \frac{-2x}{-2} > \frac{6}{-2} \] So, \( x > -3 \). * If \( \frac{y}{-4} \ge 3 \), then multiplying by \( -4 \) means we must reverse the sign: \[ \frac{y}{-4} \times (-4) \le 3 \times (-4) \] So, \( y \le -12 \). #### Representing Solutions on a Number Line The solution to a linear inequality is a set of all real numbers that make the inequality true. We can represent this solution set graphically on a number line. * **Open Circle (\( \circ \))**: Used for inequalities with \( < \) or \( > \). This means the endpoint is *not* included in the solution set. * **Closed Circle (\( \bullet \))**: Used for inequalities with \( \le \) or \( \ge \). This means the endpoint *is* included in the solution set. * **Arrow**: An arrow extends from the circle in the direction of the solution set. * For \( x > a \), the arrow points to the right. * For \( x < a \), the arrow points to the left. ### Vocabulary * **Inequality**: A mathematical statement comparing two expressions using comparison signs (\( <, >, \le, \ge \)). * **Linear Inequality**: An inequality where the highest power of the variable is 1. * **Solution Set**: The range of values for the variable that makes the inequality true. * **Number Line**: A visual representation of numbers in order, used to graph solution sets of inequalities. * **Open Circle**: A circle (\( \circ \)) on a number line indicating that an endpoint is not included in the solution. * **Closed Circle**: A filled circle (\( \bullet \)) on a number line indicating that an endpoint is included in the solution. ### Examples #### Example 1: Basic Addition Solve \( x + 5 > 9 \) and represent the solution on a number line. **Solution:** \( x + 5 > 9 \) Subtract 5 from both sides: \( x + 5 - 5 > 9 - 5 \) \( x > 4 \) On a number line, this would be an open circle at 4, with an arrow pointing to the right. #### Example 2: Basic Multiplication Solve \( 3x - 2 \le 10 \) and represent the solution on a number line. **Solution:** \( 3x - 2 \le 10 \) Add 2 to both sides: \( 3x - 2 + 2 \le 10 + 2 \) \( 3x \le 12 \) Divide by 3 (positive number, so no sign change): \( \frac{3x}{3} \le \frac{12}{3} \) \( x \le 4 \) On a number line, this would be a closed circle at 4, with an arrow pointing to the left. #### Example 3: Division by a Negative Number Solve \( -2x + 1 < 7 \) and represent the solution on a number line. **Solution:** \( -2x + 1 < 7 \) Subtract 1 from both sides: \( -2x + 1 - 1 < 7 - 1 \) \( -2x < 6 \) Divide by \( -2 \) (negative number, so reverse the inequality sign!): \( \frac{-2x}{-2} > \frac{6}{-2} \) \( x > -3 \) On a number line, this would be an open circle at -3, with an arrow pointing to the right. #### Example 4: Mixed Operations Solve \( \frac{x}{3} - 4 \ge -2 \) and represent the solution on a number line. **Solution:** \( \frac{x}{3} - 4 \ge -2 \) Add 4 to both sides: \( \frac{x}{3} - 4 + 4 \ge -2 + 4 \) \( \frac{x}{3} \ge 2 \) Multiply by 3 (positive number, so no sign change): \( \frac{x}{3} \times 3 \ge 2 \times 3 \) \( x \ge 6 \) On a number line, this would be a closed circle at 6, with an arrow pointing to the right. ### Exercises Solve each inequality and represent its solution on a number line. 1. \( x - 3 < 5 \) 2. \( y + 7 \ge 12 \) 3. \( 4z > 20 \) 4. \( \frac{m}{2} \le 3 \) 5. \( -5p < 15 \) 6. \( 8 - w \le 6 \) 7. \( 2x + 3 \ge 11 \) 8. \( 10 - 3k > 1 \) ### Assignment Solve the following inequalities and draw a number line to show the solution set for each. 1. \( n + 9 < 4 \) 2. \( 6g \le 30 \) 3. \( -y + 2 \ge 5 \) 4. \( \frac{a}{-3} > -2 \) 5. \( 4(x - 1) < 12 \) 6. \( 7 - 2b \ge 1 \) 7. \( 3m + 5 \le -4 \) 8. \( 15 < 2p + 3 \) ### Summary In this lesson, you have learned about linear inequalities, which are mathematical statements comparing two expressions using signs like \( <, >, \le, \ge \). You discovered that 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.** Finally, you learned how to represent the solution set of an inequality graphically on a number line, using open circles for strict inequalities (\( <, > \)) and closed circles for inclusive inequalities (\( \le, \ge \)). Understanding linear inequalities is a fundamental skill that will be used in many future mathematics topics. ## Inequalities on the number line ### Introduction Welcome to this lesson on inequalities! In mathematics, we often deal with equations where two expressions are equal. However, sometimes we need to compare quantities that are not equal. This is where inequalities come in. An inequality is a mathematical statement that shows the relationship between two expressions that are not necessarily equal. In this lesson, we will learn how to write, understand, and represent these inequalities using a visual tool called the number line. ### Key Concepts #### What is an Inequality? An inequality is a statement that one quantity is greater than, less than, greater than or equal to, or less than or equal to another quantity. Unlike an equation which uses an equals sign \(=\), an inequality uses special symbols to show comparison. #### Inequality Symbols There are four main inequality symbols you need to know: * **\( < \)** : Less than * Example: \( 3 < 5 \) (3 is less than 5) * **\( > \)** : Greater than * Example: \( 7 > 2 \) (7 is greater than 2) * **\( \le \)** : Less than or equal to * Example: \( x \le 10 \) (x is less than or equal to 10). This means x can be 10 or any number smaller than 10. * **\( \ge \)** : Greater than or equal to * Example: \( y \ge -4 \) (y is greater than or equal to -4). This means y can be -4 or any number larger than -4. #### Representing Inequalities on a Number Line The number line is a powerful visual tool to represent inequalities. It helps us see all the possible values that satisfy an inequality. ##### Open Circle vs. Closed Circle When we plot an inequality on a number line, we use circles at the specific number indicated in the inequality: * **Open Circle ( \( \circ \) )**: Used for 'less than' \( < \) and 'greater than' \( > \). This indicates that the number itself is *not* included in the solution set. * Think of it as the value being just on the edge, but not actually part of the solution. * **Closed Circle ( \( \bullet \) )**: Used for 'less than or equal to' \( \le \) and 'greater than or equal to' \( \ge \). This indicates that the number itself *is* included in the solution set. * Think of it as the value being part of the solution. ##### Shading Direction After placing the correct circle, we shade the number line in the direction of the values that satisfy the inequality: * **For \( < \) or \( \le \)**: Shade to the **left** of the circle, as these are smaller values. * **For \( > \) or \( \ge \)**: Shade to the **right** of the circle, as these are larger values. **General steps to represent an inequality on a number line:** 1. Draw a number line and mark the key number (the number in the inequality, e.g., 5 in \( x > 5 \)). 2. Decide whether to use an open or closed circle at that number based on the inequality symbol. 3. Shade the number line in the direction that represents the solution set. #### Compound Inequalities Sometimes, an unknown value might satisfy two inequalities at the same time. These are called compound inequalities. For example, \( -2 < x \le 3 \) means that \( x \) is greater than -2 AND \( x \) is less than or equal to 3. On a number line, this would be a segment between -2 and 3, with an open circle at -2 and a closed circle at 3. ### Vocabulary * **Inequality**: A mathematical statement that compares two values, showing that one is less than, greater than, less than or equal to, or greater than or equal to another. * **Number Line**: A straight line on which numbers are marked at regular intervals, used to visually represent inequalities. * **Less Than (\( < \))**: Indicates that one value is smaller than another. * **Greater Than (\( > \))**: Indicates that one value is larger than another. * **Less Than or Equal To (\( \le \))**: Indicates that one value is smaller than or the same as another. * **Greater Than or Equal To (\( \ge \))**: Indicates that one value is larger than or the same as another. * **Solution Set**: The set of all values that make an inequality true. * **Open Circle**: A circle \( \circ \) used on a number line to show that an endpoint is *not* included in the solution. * **Closed Circle**: A circle \( \bullet \) used on a number line to show that an endpoint *is* included in the solution. * **Compound Inequality**: An inequality that combines two simple inequalities, often showing that a value is between two other values. ### Examples **Example 1: Representing \( x > 4 \) on a number line.** 1. Draw a number line. Mark 4 and a few numbers around it (e.g., 2, 3, 4, 5, 6). 2. Since it's \( > \), place an **open circle** at 4. 3. Since \( x \) is *greater than* 4, shade the line to the **right** of 4. **Example 2: Representing \( y \le -1 \) on a number line.** 1. Draw a number line. Mark -1 and a few numbers around it (e.g., -3, -2, -1, 0, 1). 2. Since it's \( \le \), place a **closed circle** at -1. 3. Since \( y \) is *less than or equal to* -1, shade the line to the **left** of -1. **Example 3: Representing "A number \( p \) is at least 6" as an inequality and on a number line.** 1. "At least 6" means the number can be 6 or greater than 6. So, \( p \ge 6 \). 2. On a number line, place a **closed circle** at 6. 3. Shade the line to the **right** of 6. **Example 4: Representing "The temperature \( T \) is between \( 10^{\circ}C \) and \( 20^{\circ}C \) (not including \( 10^{\circ}C \) but including \( 20^{\circ}C \))" as an inequality and on a number line.** 1. This is a compound inequality: \( 10 < T \le 20 \). 2. On a number line, place an **open circle** at 10 and a **closed circle** at 20. 3. Shade the region **between** 10 and 20. ### Exercises 1. Write the inequality for each statement: * a) A number \( k \) is less than 12. * b) The speed \( s \) must be greater than 80 km/h. * c) The height \( h \) of a person must be at most 180 cm. * d) The age \( a \) of a child is between 5 and 10 years, including 5 but not including 10. 2. Represent each inequality on a number line: * a) \( x > 2 \) * b) \( m \le 0 \) * c) \( y < -3 \) * d) \( p \ge 5 \) * e) \( -1 < z \le 4 \) 3. Write the inequality shown by each number line graph: * a) An open circle at 7, shaded to the right. * b) A closed circle at -2, shaded to the left. * c) An open circle at -5, a closed circle at 1, with the region between them shaded. ### Assignment 1. Explain in your own words the difference between an open circle and a closed circle when graphing inequalities on a number line. 2. For each scenario, write an inequality and draw its representation on a number line: * a) You must be at least 16 years old to get a driving permit. * b) The maximum capacity of an elevator is 10 people. * c) The temperature in a room should be no less than \( 20^{\circ}C \) and no more than \( 25^{\circ}C \). * d) To qualify for a discount, a customer must spend more than \( \$50 \). 3. Identify the error in the following number line representation for \( x \ge -4 \) and draw the correct representation: * (Imagine a number line with an open circle at -4, shaded to the right.) ### Summary In this lesson, we explored inequalities and how to represent them visually on a number line. We learned that inequalities use symbols like \( < \) (less than), \( > \) (greater than), \( \le \) (less than or equal to), and \( \ge \) (greater than or equal to) to compare quantities. To graph inequalities, we use an **open circle** (\( \circ \)) for \( < \) or \( > \) to show that the endpoint is not included, and a **closed circle** (\( \bullet \)) for \( \le \) or \( \ge \) to show that the endpoint is included. The direction of shading on the number line indicates all the possible values that satisfy the inequality. Understanding inequalities is a fundamental skill in mathematics that helps us describe a range of possibilities rather than just a single exact value.