Question

solve the following absolute value inequality.
\frac{|x - 3|}{2} > 6
x > ? or x <

Explanation:

Step1: Eliminate the denominator

Multiply both sides of the inequality \(\frac{|x - 3|}{2}>6\) by 2 to get \(|x - 3|>12\).

Step2: Solve the absolute - value inequality

For an absolute - value inequality \(|a|>b\) (where \(b>0\)), it is equivalent to \(a > b\) or \(a < -b\).
Here, \(a=x - 3\) and \(b = 12\), so we have two cases:

Case 1: \(x-3>12\)

Add 3 to both sides of the inequality \(x - 3>12\), we get \(x>12 + 3=15\).

Case 2: \(x - 3<-12\)

Add 3 to both sides of the inequality \(x - 3<-12\), we get \(x<-12 + 3=-9\).

Answer:

For \(x>\square\), the value is \(15\); for \(x<\square\), the value is \(- 9\). So \(x>15\) or \(x < - 9\).