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Chapter 0: Problem 61

Solve each absolute value inequality. $$|x+3| \leq 4$$

Short Answer

Expert verified
The solution set is \(x \in [-1, 1]\).

Step by step solution

01

Resolve Absolute Value

The original inequality can be rewritten as two separate inequalities: \(x + 3 \leq 4\) and \(-(x + 3) \leq 4\). These inequalities cover the two possibilities that the absolute value allows for, and both must be solved independently.
02

Solve First Inequality

Subtract 3 from both sides of the first inequality \(x+3 \leq 4\) to isolate \(x\). This leads to the solution \(x \leq 1\).
03

Solve Second Inequality

Subtract 3 from both sides of the second inequality \(-(x+3) \leq 4\). This leads to \(-x \leq 1\). Multiply both sides by \(-1\) to solve for \(x\). When multiplying or dividing an inequality by a negative number, the direction of the inequality changes, leading to \(x \geq -1\).
04

Combine Results

The results of both inequalities are combined to form the final solution. Therefore, the solution to the inequality \(|x+3| \leq 4\) is \(x \in [-1, 1]\) because it satisfies both \(x \leq 1\) and \(x \geq -1\).

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