Question

1. |4x + 1| > 7

Explanation:

Step1: Recall absolute value inequality rule

For \(|A| > B\) (where \(B>0\)), it is equivalent to \(A > B\) or \(A < -B\). Here, \(A = 4x + 1\) and \(B = 7\). So we have two inequalities:
\(4x + 1 > 7\) or \(4x + 1 < -7\)

Step2: Solve \(4x + 1 > 7\)

Subtract 1 from both sides: \(4x + 1 - 1 > 7 - 1\)
Simplify: \(4x > 6\)
Divide both sides by 4: \(x > \frac{6}{4}=\frac{3}{2}\)

Step3: Solve \(4x + 1 < -7\)

Subtract 1 from both sides: \(4x + 1 - 1 < -7 - 1\)
Simplify: \(4x < -8\)
Divide both sides by 4: \(x < \frac{-8}{4}=-2\)

Answer:

The solution to the inequality \(|4x + 1| > 7\) is \(x < -2\) or \(x > \frac{3}{2}\)