Question

solve the following absolute value inequality.\\(\frac{2|x - 1|}{5} geq 2\\)\\(x geq ?\\) or \\(x leq \square\\)

Explanation:

Step1: Eliminate the denominator

Multiply both sides of the inequality \(\frac{2|x - 1|}{5}\geq2\) by \(5\) to get \(2|x - 1|\geq10\).

Step2: Simplify the coefficient of absolute value

Divide both sides of the inequality \(2|x - 1|\geq10\) by \(2\) to obtain \(|x - 1|\geq5\).

Step3: Solve the absolute - value inequality

The absolute - value inequality \(|a|\geq b\) (where \(b>0\)) is equivalent to \(a\geq b\) or \(a\leq - b\). For \(|x - 1|\geq5\), we have two cases:

• Case 1: \(x - 1\geq5\)

Add \(1\) to both sides of the inequality \(x - 1\geq5\), we get \(x\geq5 + 1=6\).

• Case 2: \(x - 1\leq - 5\)

Add \(1\) to both sides of the inequality \(x - 1\leq - 5\), we get \(x\leq-5 + 1=-4\).

Answer:

\(x\geq\boldsymbol{6}\) or \(x\leq\boldsymbol{-4}\)