Question

Solve the inequality. Then graph the solution. $$|2 x+5|-1<6$$

Short answer

The solution of the inequality is \(-5<x<1\).

Step-by-step solution

Setup Two Inequalities

First of all, set up two inequalities from the given absolute value inequality. The first one is \(2x+5-1<6\) and the second one is \(-(2x+5-1)<6\). This is because the absolute value \(\mid a \mid\) can be thought of as the number \(a\) or \(-a\), depending on the sign of \(a\).

Simplify The Inequalities

Simplify both inequalities to isolate \(x\). For the first inequality \(2x+5-1<6\), simplify it to \(2x+4<6\) and then \(2x<2\) which simplifies to \(x<1\). For the second inequality, simplify \(-(2x+5-1)<6\) to \(-2x-4<6\) which simplifies to \(-2x<10\) and then \(x>-5\) by dividing both sides by -2 and flipping the sign of the inequality.

Graph the Solution

The solution to the inequality is where these two solutions overlap which is \(-5<x<1\). This can be graphed on a number line. Place two open circles at -5 and 1, and shade the region between these two points. An open circle is used because the original inequality does not include 'equal to' so the end points -5 and 1 are not a part of the solution.