# How do you solve  |.5x+9|–x+3<0?

Jan 23, 2018

Break the $x$-axis into parts, then we can find that only one range of $x$ solves the inequality: $x > 24$

#### Explanation:

The simplest way to solve problems involving absolute values is to break the domain ($x$-axis) into parts, those where the result inside the absolute value is positive, and those where the result is negative. Then we can explicitly make it positive by multiplying by $+ 1$ or $- 1$ thus removing the absolute value. We can then solve normally.

In this case we need to know where $0.5 x + 9$ is positive:

$0.5 x + 9 \ge 0$

$0.5 x \ge - 9$

$x \ge - 4.5$

And by inspection, it is negative for $x < - 4.5$. We now have our two domains. Starting with the negative:

|0.5x+9|–x+3<0

 -1*(0.5x+9)–x+3<0 for $x < - 4.5$

$\to - 0.5 x - 9 - x + 3 < 0$

$\to - 1.5 x - 6 < 0$

note that when we multiply or divide by a negative value in this step, we must change the direction of the inequality:

$\to x + 4 > 0$

$\to x > - 4$ which cannot happen in the domain $x < - 4.5$ therefore there is no solution here.

Now for the positive part of the absolute value where $x \ge - 4.5$

 0.5x+9–x+3<0 for $x \ge - 4.5$

$\to - 0.5 x + 12 < 0$

$\to - x < - 24$

$\to x > 24$ which is in the domain $x \ge - 4.5$. So the solution is

$x > 24$