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Chapter 1: Problem 82

Solve each absolute value inequality. $$-3|x+7| \geq-27$$

Short Answer

Expert verified
The solution to the inequality \(-3|x+7| \geq-27\) is \([-16, 2]\).

Step by step solution

01

Isolate the Absolute Value

The goal is to isolate the absolute value on one side of the inequality. To do this, divide both sides of the inequality by -3. But remember, dividing or multiplying both sides of an inequality by a negative number will flip the inequality symbol direction. Consequently, \(|x+7| \leq 9\).
02

Remove the Absolute Value

The rule is for any number \(a\), \(|a| = a\) if \(a \geq 0\) and \(|a| = -a\) if \(a < 0\). So, we create two inequalities to represent both conditions, \(x + 7 \leq 9\) and \(-x - 7 \leq 9\).
03

Solve Each Inequality

Next, solve the two inequalities: for \(x + 7 \leq 9\), subtract 7 from both sides to get \(x \leq 2\); for \(-x - 7 \leq 9\), add 7 to both sides to get \(-x \leq 16\), then multiply both sides by -1 to get \(x \geq -16\). Remember, when multiplying both sides by a negative number, the inequality symbol flips.
04

Interpret the Solution

The solution to both parts of the compound inequality (-- the 'and' inequality \(x \geq -16\) and the 'or' inequality \(x \leq 2\) -- is the overlapping interval between the solutions of each part. In this case, our solution is the interval \([-16, 2]\).

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