Question

solve the absolute value inequality
|4x - 4| < 16

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set in interval notation is
(simplify your answer.)
b. the solution set is ∅.

Explanation:

Step1: Apply absolute value inequality rule

For \(|a| < b\) (where \(b>0\)), it is equivalent to \(-b < a < b\). So for \(|4x - 4| < 16\), we have \(-16 < 4x - 4 < 16\).

Step2: Solve the left - hand inequality

Add 4 to both sides of \(-16 < 4x - 4\): \(-16+4 < 4x-4 + 4\), which simplifies to \(-12 < 4x\). Then divide both sides by 4: \(\frac{-12}{4}<\frac{4x}{4}\), so \(-3 < x\).

Step3: Solve the right - hand inequality

Add 4 to both sides of \(4x - 4 < 16\): \(4x-4 + 4<16 + 4\), which simplifies to \(4x < 20\). Then divide both sides by 4: \(\frac{4x}{4}<\frac{20}{4}\), so \(x < 5\).

Step4: Combine the inequalities

From Step 2 and Step 3, we have \(-3 < x < 5\). In interval notation, this is \((-3,5)\).

Answer:

A. The solution set in interval notation is \((-3, 5)\)