To say that \( \left|\frac{x-8}{4}\right| \leq 5 \) is the same as saying \( x \) is in the closed interval \( [A, B] \) Solve the absolute value inequality and state your answer in interval notation: Question Help: Video Submit Question

To solve the absolute value inequality: \[ \left|\frac{x - 8}{4}\right| \leq 5 \] and express the solution in interval notation \([A, B]\), follow these steps: ### Step 1: Understand the Absolute Value Inequality The inequality \(\left| \frac{x - 8}{4} \right| \leq 5\) means that the expression inside the absolute value, \(\frac{x - 8}{4}\), is **at most** 5 units away from 0 on the number line. In general, an absolute value inequality of the form \(|A| \leq B\) (where \(B \geq 0\)) can be rewritten without the absolute value as: \[ -B \leq A \leq B \] ### Step 2: Rewrite the Inequality Without Absolute Value Applying this to your inequality: \[ \left| \frac{x - 8}{4} \right| \leq 5 \quad \Rightarrow \quad -5 \leq \frac{x - 8}{4} \leq 5 \] ### Step 3: Solve for \(x\) Now, solve the compound inequality for \(x\): 1. **Multiply all parts by 4** (since 4 is positive, the direction of the inequalities remains the same): \[ -5 \times 4 \leq x - 8 \leq 5 \times 4 \] \[ -20 \leq x - 8 \leq 20 \] 2. **Add 8 to all parts** to isolate \(x\): \[ -20 + 8 \leq x \leq 20 + 8 \] \[ -12 \leq x \leq 28 \] ### Step 4: Express the Solution in Interval Notation The solution \( -12 \leq x \leq 28 \) means that \( x \) includes all real numbers from \(-12\) to \(28\), inclusive. In interval notation, this is written as: \[ [-12,\ 28] \] ### Final Answer \[ \boxed{\,[-12,\ 28]\,} \]