Problem 82 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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, \(|x+7| \leq 9\).

Step 2: 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 \(-x - 7 \leq 9\).

Step 3: 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.

Step 4: 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 \([-16, 2]\).