Question

which of the following number lines represents the solution set of the inequality | - 4x + 4| ≤ 16?

Explanation:

Step1: Remove absolute - value

For \(| - 4x + 4|\leq16\), we have \(-16\leq - 4x + 4\leq16\).

Step2: Solve the left - hand side of the compound inequality

\(-16\leq - 4x + 4\), subtract 4 from both sides: \(-16 - 4\leq - 4x\), so \(-20\leq - 4x\). Divide both sides by \(-4\) and reverse the inequality sign, we get \(x\leq5\).

Step3: Solve the right - hand side of the compound inequality

\(-4x + 4\leq16\), subtract 4 from both sides: \(-4x\leq16 - 4\), so \(-4x\leq12\). Divide both sides by \(-4\) and reverse the inequality sign, we get \(x\geq - 3\).

Answer:

The solution set is \(-3\leq x\leq5\), which is represented by a number line with closed - circles at \(x=-3\) and \(x = 5\) and the line segment between them. (The first number line shown in the image where the closed - circles are at \(-3\) and \(5\))