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GRAPHING INEQUALITIES Exploration One way to solve the double inequality −4 < 3𝑥 + 2 < 5 is: Add −2 to both sides: −6 < 3𝑥 < 3 1 Multiply both sides by 3: −2 < 𝑥 < 1 Another way is to rewrite the double inequality as two distinct inequalities: −4 < 3𝑥 + 2 and 3𝑥 + 2 < 5. Solve them separately: −6 < 3𝑥 and 3𝑥 < 3 −2 < 𝑥 and 𝑥 < 1 Combine the solution sets: −2 < 𝑥 < 1 1. Solve the double inequality −9 < 5 − 7𝑥 ≤ 12 in two ways algebraically. 2. Solve −14 < −7(3𝑥 + 2) < 21 a. By first using equivalence transformations b. By expanding the brackets first. Explain which method you prefer. 3. Does the pair of inequalities 4𝑥 < 4 and 3𝑥 − 5 ≥ 1 have a common solution set? Explain. 4. Given that −1 < 𝑥 < 4, find two values 𝑚 and 𝑛 such that 𝑚 < 2𝑥 + 3 < 𝑛. SOLVING GRAPHICALLY Let us solve the inequality 3𝑥 − 7 < 5. Draw the lines 𝑓1 (𝑥) = 3𝑥 − 7 and 𝑓2 (𝑥) = 5 and find the intersection point. The symbol < means that 3𝑥 − 7 cannot take the value 5. Use a dashed line for 𝑓2 (𝑥) = 5 to show that points on the line 𝑓2 (𝑥) = 5 are not possible solutions. The lines intersect at 𝑥 = 4. This is the value of 𝑥 that makes both sides equal. For 3𝑥 − 7 < 5 you need values of 𝑥 such that 𝑓1 (𝑥) < 𝑓2 (𝑥). From the graph, the line for 𝑓1 (𝑥) is below 𝑓2 (𝑥) when 𝑥 < 4, showing that the values of 𝑓1 (𝑥) are less than the values of 𝑓2 (𝑥) when 𝑥 < 4. The solution is 𝑥 < 4. You can check your solution by choosing a point with 𝑥 < 4, for example 𝑥 = 2. Substitute 𝑥 = 2 into the inequality: 3𝑥 − 7 ⟶ 3 × 2 − 7 = 6 − 7 = −1 < 5 Now choose a point with 𝑥 > 4, say: 𝑥 = 6. Substitute 𝑥 = 6 into the inequality: 3𝑥 − 7 ⟶ 3 × 6 − 7 = 18 − 7 = 11. 11 is not less than 5, so the point does not satisfy the inequality. CHALLENGE Solve the inequality 5𝑥 − 7 ≥ 3𝑥 + 9 graphically and then check your answer algebraically. Practice 2 Solve the following inequalities graphically: 1. 3𝑥 − 7 < 5 𝑥 2. 6 − 7 > 2 4. 5𝑥 − 7 ≥ 3𝑥 + 9 3 5. 1 − 𝑥 ≤ 𝑥 − 4 3. 3(4 − 𝑚) > 9 6. 2 2𝑥+1 − 3 > 5𝑥 + 2 7. −2(𝑎 − 3) < 5(𝑎 − 2) − 12 8. 2(1 − 𝑏) + 5 ≥ 3(2𝑏 − 1) 9. 4𝑘 − 11 ≤ 3𝑘 2 +5 Practice 3 Write down the two inequalities that could be represented by the non-shaded region of this graph. Practice 4 Verify, graphically and algebraically, that the inequality 3𝑥 + 2 < 9𝑥 + 6 has an infinite number of solutions. Which of the methods do you prefer? Explain why.