INEQUALITIES Introduction to solving simple one-step inequalities. Grade 8 Mathematics: Inequalities — Introduction to Solving Simple One-Step Inequalities — Part A Subtopic Navigator Introduction Meaning of Inequalities Inequality Symbols Solving by Addition & Subtraction Solving by Multiplication & Division Important Rule (Sign Reversal) Applications Cumulative Exercises Conclusion Introduction An inequality shows that two expressions are not necessarily equal but one is larger or smaller than the other. In this lesson, you will learn to solve simple one-step inequalities involving addition, subtraction, multiplication, and division. We will also explore the special rule: when multiplying or dividing by a negative number, the inequality sign is reversed. Meaning of Inequalities An inequality compares two quantities. For example: [latex] 7 > 3 [/latex], [latex] -2 < 5 [/latex]. Example 1 Which of the following is true: [latex]12 < 15[/latex] or [latex]12 > 15[/latex]? Solution: [latex]12 < 15[/latex] is true since 12 is less than 15. Exercises (Meaning of Inequalities) State whether [latex]-8 > -3[/latex] is true or false. Compare [latex]5[/latex] and [latex]0[/latex] using an inequality sign. Compare [latex]2.7[/latex] and [latex]2.71[/latex]. Inequality Symbols The common inequality symbols are: [latex]<[/latex] : less than [latex]>[/latex] : greater than [latex]leq[/latex] : less than or equal to [latex]geq[/latex] : greater than or equal to Example 2 Write "x is at least 10" using an inequality symbol. Solution: "At least 10" means [latex]x geq 10[/latex]. Exercises (Inequality Symbols) Write "y is not more than 4" using an inequality. Write "k is greater than -2" as an inequality. Write "m is at most 7" as an inequality. Solving by Addition & Subtraction We can solve inequalities by adding or subtracting the same number on both sides, just like equations. Example 3 Solve [latex]x + 5 < 12[/latex]. Solution: Subtract 5 from both sides: [latex]x < 12 - 5 = 7[/latex]. Exercises (Addition & Subtraction) Solve [latex]x - 3 leq 10[/latex]. Solve [latex]y + 7 > 2[/latex]. Solve [latex]k - 12 geq -5[/latex]. Solving by Multiplication & Division We can also solve inequalities by multiplying or dividing both sides by the same positive number. Example 4 Solve [latex]3x > 12[/latex]. Solution: Divide both sides by 3: [latex]x > 4[/latex]. Exercises (Multiplication & Division) Solve [latex]2y leq 14[/latex]. Solve [latex]dfrac{x}{5} geq 6[/latex]. Solve [latex]4k < 20[/latex]. Important Rule (Sign Reversal) When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign. Example 5 Solve [latex]-2x > 8[/latex]. Solution: Divide both sides by -2 (reverse sign): [latex]x < -4[/latex]. Exercises (Sign Reversal) Solve [latex]-3x < 9[/latex]. Solve [latex]-4y geq -20[/latex]. Solve [latex]dfrac{-k}{2} > 7[/latex]. Applications & Mixed Problems Inequalities are often used in real life to represent limits, ranges, and conditions. For example, when a bus can carry at most 50 passengers, we write [latex]p leq 50[/latex], where [latex]p[/latex] is the number of passengers. Example 6 A school library allows students to borrow not more than 6 books. Write this as an inequality involving [latex]b[/latex], the number of books a student borrows. Solution: Since borrowing is not more than 6: [latex]b leq 6[/latex]. Example 7 A taxi charges a base fare of ₦500 plus ₦100 per kilometre. If a passenger has at most ₦1200 to spend, write and solve an inequality to find the maximum kilometres [latex]k[/latex] the passenger can travel. Solution: Total fare: [latex]500 + 100k leq 1200[/latex]. Subtract 500: [latex]100k leq 700[/latex]. Divide by 100: [latex]k leq 7[/latex]. So the passenger can travel at most 7 km. Cumulative Exercises Solve the following inequalities. Show all working. [latex]2x + 5 < 15[/latex] [latex]3y - 7 geq 8[/latex] [latex]dfrac{k}{4} + 6 leq 3[/latex] [latex]-2x > 10[/latex] [latex]5 - 3y leq -7[/latex] [latex]-4m + 9 geq 1[/latex] [latex]dfrac{-n}{3} < 4[/latex] [latex]7p - 2 > 4p + 10[/latex] [latex]2(q - 5) leq 3q - 11[/latex] [latex]dfrac{5r - 4}{2} geq 6[/latex] Show/Hide Answers [latex]2x + 5 < 15 ;;Rightarrow;; x < 5[/latex] [latex]3y - 7 geq 8 ;;Rightarrow;; y geq 5[/latex] [latex]dfrac{k}{4} + 6 leq 3 ;;Rightarrow;; k leq -12[/latex] [latex]-2x > 10 ;;Rightarrow;; x < -5[/latex] (sign reverses) [latex]5 - 3y leq -7 ;;Rightarrow;; y geq 4[/latex] [latex]-4m + 9 geq 1 ;;Rightarrow;; m leq 2[/latex] [latex]dfrac{-n}{3} < 4 ;;Rightarrow;; n > -12[/latex] [latex]7p - 2 > 4p + 10 ;;Rightarrow;; p > 4[/latex] [latex]2(q - 5) leq 3q - 11 ;;Rightarrow;; q geq 1[/latex] [latex]dfrac{5r - 4}{2} geq 6 ;;Rightarrow;; r geq dfrac{16}{5}[/latex] Visualizing Inequalities The graph below shows shaded solutions of inequalities on a number line. For example, try typing [latex]x geq 5[/latex] or [latex]x < -2[/latex] to see how solutions appear. Conclusion In this lesson, we learned the meaning of inequalities, the common symbols used, and the rules for solving one-step inequalities. Key points include: Addition and subtraction work the same as with equations. Multiplication and division also follow equation rules, except when multiplying/dividing by a negative number — the inequality sign is reversed. Inequalities are useful in describing conditions, limits, and real-life constraints. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c
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