Question

which of the following is the solution to |x - 13| < 18?
a. x < 31 or x > -5
b. x < 31 and x > -5
c. x < 31
d. x < 31 and x < -5

Explanation:

Step1: Recall absolute value inequality rule

For \(|a| < b\) (where \(b>0\)), it is equivalent to \(-b < a < b\). So for \(|x - 13| < 18\), we can rewrite it as \(-18 < x - 13 < 18\).

Step2: Solve the compound inequality

First, solve the left - hand side of the inequality \(-18 < x - 13\). Add 13 to both sides: \(-18+13 < x-13 + 13\), which simplifies to \(-5 < x\).
Then, solve the right - hand side of the inequality \(x - 13 < 18\). Add 13 to both sides: \(x-13 + 13<18 + 13\), which simplifies to \(x < 31\).

Combining these two results, we get \(-5 < x\) and \(x < 31\), or in other words, \(x < 31\) and \(x>-5\).

Answer:

B. \(x < 31\) and \(x > - 5\)