Question

Solve the inequality. $$ 6<4 x-2 \leq 14 $$

Short answer

The solution to the inequality \(6 < 4x - 2 \leq 14\) is \(2 < x \leq 4\).

Step-by-step solution

Isolate the inequality

The first step is to isolate the compound inequality. The original inequality \(6 < 4x - 2 \leq 14\) can be split into two separate inequalities: \(6 < 4x - 2\) and \(4x - 2 \leq 14\).

Solve the first inequality

To solve the inequality \(6 < 4x - 2\), first add 2 to both sides to remove the -2 on the right side of the inequality. This results in \(8 < 4x\). Next, divide both sides by 4 to solve for \(x\), yielding \(x > 2\).

Solve the second inequality

Likewise, for the inequality \(4x - 2 \leq 14\), add 2 to both sides to get \(4x \leq 16\). After doing so, divide both sides by 4 to solve for \(x\), yielding \(x \leq 4\).

Combine the solutions

The solution to the original compound inequality is the intersection of the solutions to the two individual inequalities. Therefore, the compound inequality \(6 < 4x - 2 \leq 14\) has the solution \(2 < x \leq 4\).