Question

1. solve the inequality: |x + 9| ≥ 14

a. -2 < x < 8
b. x ≥ 5 or x ≤ -23
c. x > 6 or x < -6
d. x ≥ 6 or x ≤ -16

Explanation:

Step1: Recall absolute value inequality rule

For \(|a| \geq b\) (where \(b>0\)), it is equivalent to \(a \geq b\) or \(a \leq -b\). So for \(|x + 9| \geq 14\), we have two cases:
Case 1: \(x + 9 \geq 14\)
Case 2: \(x + 9 \leq - 14\)

Step2: Solve Case 1: \(x + 9 \geq 14\)

Subtract 9 from both sides: \(x+9 - 9\geq14 - 9\)
Simplify: \(x \geq 5\)

Step3: Solve Case 2: \(x + 9 \leq - 14\)

Subtract 9 from both sides: \(x+9 - 9\leq - 14 - 9\)
Simplify: \(x \leq - 23\)

Answer:

B. \(x \geq 5\) or \(x \leq -23\)