Question

1. solve the inequality: |r| < 4

a. -4 < r < 4
b. -4 ≤ r ≤ 4
c. -6 < r < 2
d. r < -6 or r > 2

1. solve the inequality: |x + 9| ≥ 14

a. -2 < x < 8
b. x ≥ 5 or x ≤ -23

Explanation:

Question 12

Step1: Recall absolute value inequality rule

For \(|a| < b\) (where \(b>0\)), it is equivalent to \(-b < a < b\).

Step2: Apply the rule to \(|r| < 4\)

Here, \(a = r\) and \(b = 4\). So applying the rule, we get \(-4 < r < 4\).

Step1: Recall absolute value inequality rule

For \(|a| \geq b\) (where \(b>0\)), it is equivalent to \(a \geq b\) or \(a \leq -b\).

Step2: Apply the rule to \(|x + 9| \geq 14\)

Here, \(a = x + 9\) and \(b = 14\). So we have two cases:
Case 1: \(x + 9 \geq 14\)
Subtract 9 from both sides: \(x \geq 14 - 9 = 5\)
Case 2: \(x + 9 \leq -14\)
Subtract 9 from both sides: \(x \leq -14 - 9 = -23\)
Combining both cases, we get \(x \geq 5\) or \(x \leq -23\).

Answer:

A. \(-4 < r < 4\)

Question 13