Question

Solve the inequality. Then graph the solution. $$|x+12|>36$$

Short answer

The solution to the inequality \(|x+12|>36\) is \(x < -48\) or \(x > 24\).

Step-by-step solution

Break down the absolute inequality into two separate inequalities

When dealing with absolute value inequalities, it's best to convert them into two separate inequalities. Absolute value inequalities involve values that are greater or less than the absolute value or the number's distance from zero. Thus, the inequality can be written as \(x + 12 > 36\) and \(x + 12 < -36\).

Solve each inequality separately

Now, solve each inequality separately. Substract 12 from both sides of each inequality. For the first inequality, \(x > 24\). For the second inequality, \(x < -48\).

Plot the solutions on the number line

Plot these solutions on a number line. Open circles are used at \(x = -48\) and \(x = 24\) to show that these values are not included in the solutions. The number line will have a part pointing to the left from -48 and a part pointing to the right from 24.