Question

solve the following absolute value inequality.\\(\frac{|x - 3|}{2} > 1\\)\\(x > ?\\) or \\(x <\\)

Explanation:

Step1: Eliminate the denominator

Multiply both sides of the inequality \(\frac{|x - 3|}{2}>1\) by 2 to get \(|x - 3|>2\).

Step2: Solve the absolute - value inequality

The absolute - value inequality \(|a|>b\) (where \(b > 0\)) is equivalent to \(a>b\) or \(a < -b\).
For \(|x - 3|>2\), we have two cases:
Case 1: \(x - 3>2\)
Add 3 to both sides of the inequality \(x - 3>2\), we get \(x>2 + 3\), so \(x>5\).
Case 2: \(x - 3<-2\)
Add 3 to both sides of the inequality \(x - 3<-2\), we get \(x<-2 + 3\), so \(x < 1\).

Answer:

\(x>5\) or \(x < 1\) (The value in the square bracket is 5)