Question

1. solve the inequality: $|-2x| < 14$

a. $-4 \leq x \leq 4$
b. $0 < x \leq 4$
c. $x < -7$ or $x > 7$
d. $-7 < x < 7$

Explanation:

Step1: Recall absolute value inequality rule

For \(|a| < b\) (where \(b>0\)), it is equivalent to \(-b < a < b\). Also, \(|-2x| = |2x|\) because absolute value of a negative number is its positive counterpart. So we can rewrite \(|-2x| < 14\) as \(|2x| < 14\), and then apply the absolute value inequality rule: \(-14 < 2x < 14\).

Step2: Solve for \(x\)

Divide each part of the compound inequality by 2.
For the left part: \(\frac{-14}{2}<\frac{2x}{2}\), which simplifies to \(-7 < x\).
For the right part: \(\frac{2x}{2}<\frac{14}{2}\), which simplifies to \(x < 7\).
Combining these, we get \(-7 < x < 7\).

Answer:

D. \(-7 < x < 7\)