Question

Solve the inequality. $$|4 x+2|-1<5$$

Short answer

The solution to the inequality is \(-0.5 < x < 1\).

Step-by-step solution

Isolate the absolute value expression

This inequality reads 'the absolute value of \(4x + 2\), subtract 1, is less than 5'. First off, isolate the absolute value expression by adding 1 on either side. This gives us a new inequality, \(|4x + 2| < 6\).

Create two inequalities

Next, remember that any absolute value inequality creates two separate inequalities. These are \(4x + 2 < 6\) and \(-(4x + 2) < 6\).

Solve the inequalities

Solve both of these inequalities. The first reduces to \(x < 1\). The second one once simplified turns into \(-4x-2<6\), which further simplifies into \(x>-0.5\).

Combine the two solutions

The final solution is the overlap between the solutions to both inequalities. Thus, the solution is \(x\) is between \(-0.5\) and \(1\).