Question

|2x - 5| ≥ 3
choose the answer that gives both the correct solution and the correct graph.
a. solution: x ≤ 1 or x ≥ 4
b. solution: x ≤ 1 or x ≥ 4
c. solution: x ≥ 1 and x ≤ 4
d. solution: x ≤ -4 or x ≥ -1

Explanation:

Step1: Split absolute - value inequality

For \(|2x - 5|\geq3\), we have two cases: \(2x-5\geq3\) or \(2x - 5\leq - 3\).

Step2: Solve \(2x-5\geq3\)

Add 5 to both sides: \(2x\geq3 + 5\), so \(2x\geq8\). Divide both sides by 2, we get \(x\geq4\).

Step3: Solve \(2x - 5\leq - 3\)

Add 5 to both sides: \(2x\leq-3 + 5\), so \(2x\leq2\). Divide both sides by 2, we get \(x\leq1\).

Answer:

A. Solution: \(x\leq1\) or \(x\geq4\)
(The graph in option A correctly represents the solution set with closed - circles at \(x = 1\) and \(x = 4\) and shading to the left of \(x = 1\) and to the right of \(x = 4\))