Inequal ities. Grade 7 Mathematics: Introduction to Solving Simple One-Step Inequalities Subtopic Navigator Introduction to Inequalities What are Inequalities? Symbols & Meaning Solving One-Step Addition/Subtraction Inequalities Solving One-Step Multiplication Inequalities The Special Case: Multiplying/Dividing by a Negative Graphing Solutions on a Number Line Cumulative Exercises Conclusion Lesson Objectives Identify and interpret the inequality symbols: >, <, ≥, ≤. Solve one-step inequalities involving addition, subtraction, and multiplication. Understand and correctly apply the rule for reversing the inequality sign when multiplying or dividing by a negative number. Represent the solution set of an inequality on a number line. Translate simple real-world situations into inequalities and solve them. Check the reasonableness of a solution to an inequality. Introduction to Inequalities What does it mean to have "at least" $5 to buy a snack, or to be "younger than" 13 to get a student discount? These everyday phrases are secretly talking about inequalities! While equations show exact equality (like x = 5), inequalities show a relationship where one side is not exactly equal to the other, but greater than, less than, or possibly equal. They help us describe ranges of possible values, like minimum heights for rides, maximum speeds on roads, or budgeting your allowance. Today, we learn the language and basic tools to solve these mathematical statements. Symbols & Meaning An inequality is a mathematical sentence that compares two expressions using an inequality symbol. It tells us that one expression is greater than, less than, or possibly equal to another. Understanding the four main symbols is our first step. Key Symbols & Meaning: [latex] > quad[/latex] means "is greater than" [latex] < quad[/latex] means "is less than" [latex] ge quad[/latex] means "is greater than or equal to" [latex] le quad[/latex] means "is less than or equal to" Example 1: Translating Words to Symbols Write the inequality for: "A number x is at most 10." Solution: "At most" means it can be 10 or anything less than 10. This is represented by "less than or equal to." [latex]x le 10[/latex] Example 2: Interpreting the Symbol What does the inequality [latex]m > -2[/latex] mean in words? Solution: The symbol `>` means "greater than." The inequality states that the variable m represents all numbers that are greater than -2. Phrase Inequality Symbol Example Inequality is more than, is above > [latex]h > 48[/latex] (height > 48 inches) is less than, is below < [latex]s < 65[/latex] (speed < 65 mph) is at least, is no less than, minimum ≥ [latex]b ge 5[/latex] (at least $5) is at most, is no more than, maximum ≤ [latex]w le 50[/latex] (weight ≤ 50 kg) Practice Problems Write an inequality for: "A number k is less than 15." Translate into words: [latex]t ge 30[/latex]. Write an inequality for: "You must be at least 16 years old to drive." Let a = age. Which symbol means "is no more than"? True or False: The number 5 is a solution to the inequality [latex]x > 5[/latex]. Addition/Subtraction Inequalities Solving an inequality means finding all possible values for the variable that make the inequality true. The process is very similar to solving equations: we use inverse operations to isolate the variable. For addition and subtraction, the rules are identical to equations. Key Principle: You can add the same number to, or subtract the same number from, both sides of an inequality, and the inequality sign remains the same. If [latex]a < b[/latex], then [latex]a + c < b + c[/latex] and [latex]a - c < b - c[/latex]. Example 1: Subtraction to Solve Solve for x: [latex]x + 7 > 12[/latex] Solution: Isolate x by subtracting 7 from both sides. [latex]x + 7 - 7 > 12 - 7[/latex] [latex]x > 5[/latex] The solution is all numbers greater than 5. Example 2: Addition to Solve Solve for y: [latex]y - 3 le -1[/latex] Solution: Isolate y by adding 3 to both sides. [latex]y - 3 + 3 le -1 + 3[/latex] [latex]y le 2[/latex] The solution is all numbers less than or equal to 2. Practice Problems Solve: [latex]n + 5 < 9[/latex] Solve: [latex]p - 2 ge 4[/latex] Solve: [latex] -6 + a > 1[/latex] Write an inequality for "7 more than z is at least 11" and solve it. Check if x = 0 is a solution to the inequality [latex]x - 4 le -3[/latex]. One-Step Multiplication Inequalities When the variable is multiplied by a positive number, we also follow the same rule as equations: divide both sides by that positive number. The inequality sign stays the same. Key Principle (Positive Multiplier/Divisor): You can multiply or divide both sides of an inequality by the same positive number, and the inequality sign remains the same. If [latex]a < b[/latex] and [latex]c > 0[/latex], then [latex]a times c < b times c[/latex] and [latex]frac{a}{c} < frac{b}{c}[/latex]. Example 1: Division by a Positive Solve for x: [latex]4x < 20[/latex] Solution: Isolate x by dividing both sides by 4 (a positive number). [latex]frac{4x}{4} < frac{20}{4}[/latex] [latex]x < 5[/latex] Example 2: Multiplication by a Positive Solve for m: [latex]frac{m}{3} ge -2[/latex] Solution: Isolate m by multiplying both sides by 3 (a positive number). [latex]3 times frac{m}{3} ge 3 times (-2)[/latex] [latex]m ge -6[/latex] Practice Problems Solve: [latex]8y > 32[/latex] Solve: [latex]frac{c}{5} le 3[/latex] Solve: [latex]12 le 2t[/latex] Three times a number k is less than 21. Write and solve the inequality. Solve: [latex]7x le 0[/latex] Multiplying/Dividing by a Negative This is the critical difference between solving equations and inequalities. When you multiply or divide both sides of an inequality by a negative number, you must reverse (flip) the inequality sign. Think about it: 2 < 4 is true. If you multiply both sides by -1, you get -2 and -4. Is -2 < -4? No! -2 is greater than -4. To keep the statement true, we must flip the sign: -2 > -4. Key Rule (Negative Multiplier/Divisor): If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. [latex] > leftrightarrow < quad text{and} quad ge leftrightarrow le [/latex] Example 1: Division by a Negative Solve for x: [latex]-3x ge 15[/latex] Solution: Isolate x by dividing both sides by -3. Because -3 is negative, we must flip the ≥ sign to ≤. [latex]frac{-3x}{-3} le frac{15}{-3}[/latex] [latex]x le -5[/latex] Example 2: Multiplication by a Negative Solve for y: [latex]frac{y}{-2} < 7[/latex] Solution: Isolate y by multiplying both sides by -2. Because -2 is negative, we must flip the < sign to >. [latex]-2 times frac{y}{-2} > -2 times 7[/latex] [latex]y > -14[/latex] Practice Problems Solve: [latex]-5a > 25[/latex] Solve: [latex]frac{n}{-4} le 2[/latex] Solve: [latex]-x ge 8[/latex] (Hint: -x means -1x) Solve: [latex]-12 < 6m[/latex]. (Careful: Isolate m first. Is your divisor positive or negative?) Why do we flip the sign when multiplying/dividing by a negative number? Explain with a simple example. Graphing Solutions on a Number Line A solution to an inequality is often not a single number, but a whole set of numbers. We can visually represent this solution set on a number line. This provides a clear picture of all possible answers. Graphing Rules: Open Circle (○): Used for > or < to show the endpoint is not included in the solution. Closed Circle (●): Used for ≤ or ≥ to show the endpoint is included. Shaded Line/Arrow: Shade the portion of the number line that contains all the solutions. Number Line Diagram: x > 3 An open circle at 3, with shading/arrow extending to the right. Example 1: Graphing x ≤ -1 1. The symbol is "≤" (less than or equal to), so we use a closed circle at -1. 2. The solutions are all numbers less than -1, so we shade the number line to the left of -1. Example 2: Graphing 2 < y (or y > 2) 1. The symbol is ">" (greater than), so we use an open circle at 2. 2. The solutions are all numbers greater than 2, so we shade the number line to the right of 2. Practice Problems Graph the solution set for [latex]x ge 0[/latex]. Graph the solution set for [latex]p < -2[/latex]. Write the inequality represented by a number line with a closed circle at 4 and shading to the left. Graph the solution to [latex]-3x le 12[/latex]. (Hint: Solve it first!) Is the circle open or closed for the inequality [latex]m > -5[/latex]? Cumulative Exercises Solve each inequality and graph its solution set on a number line where applicable. [latex]x - 8 > -3[/latex] [latex]6y le 42[/latex] [latex]frac{k}{-3} > 4[/latex] [latex] -2 + m < 5[/latex] [latex]-7p ge 0[/latex] [latex]10 ge b + 12[/latex] [latex]frac{w}{5} < -10[/latex] A student needs to score at least 70 points on a test to pass. Write an inequality for the passing score s. You have $20. Each ticket to the fair costs $4. Write and solve an inequality to find the maximum number of tickets t you can buy. Challenge: Solve and graph: [latex] 4 - x ge 9[/latex] Show/Hide Answers Problem: [latex]x - 8 > -3[/latex] Answer: Add 8 to both sides. [latex]x > 5[/latex] Problem: [latex]6y le 42[/latex] Answer: Divide both sides by 6 (positive). [latex]y le 7[/latex] Problem: [latex]frac{k}{-3} > 4[/latex] Answer: Multiply both sides by -3 (negative, flip sign). [latex]k < -12[/latex] Problem: [latex] -2 + m < 5[/latex] Answer: Add 2 to both sides. [latex]m < 7[/latex] Problem: [latex]-7p ge 0[/latex] Answer: Divide both sides by -7 (negative, flip sign). [latex]p le 0[/latex] Problem: [latex]10 ge b + 12[/latex] Answer: Subtract 12 from both sides. [latex]-2 ge b[/latex] or [latex]b le -2[/latex] Problem: [latex]frac{w}{5} < -10[/latex] Answer: Multiply both sides by 5 (positive). [latex]w < -50[/latex] Problem: Passing score s is at least 70. Answer: [latex]s ge 70[/latex] Problem: Tickets cost $4 each, maximum spend $20. Answer: Inequality: [latex]4t le 20[/latex]. Solve: [latex]t le 5[/latex]. You can buy at most 5 tickets. Problem: [latex] 4 - x ge 9[/latex] Answer: Subtract 4: [latex]-x ge 5[/latex]. Multiply by -1 (flip sign): [latex]x le -5[/latex]. Graph: Closed circle at -5, shading left. Conclusion/Recap Congratulations! You've taken your first steps into the world of inequalities. You've learned to interpret the symbols, solve one-step inequalities using addition, subtraction, and multiplication, and you've mastered the crucial rule of flipping the inequality sign when multiplying or dividing by a negative number. Remember, the solution to an inequality is often a range of numbers, beautifully represented on a number line with open or closed circles. This foundational skill is essential for algebra, as you'll soon use it to solve more complex, multi-step inequalities and apply them to real-world problems involving limits, budgets, measurements, and conditions. Clip It! Share your ANSWER in the Chat. 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