# How do you solve abs(3x-9)-2<=7?

Jul 22, 2016

$x \in \left[0 , 6\right]$

#### Explanation:

The first thing to do here is isolate the modulus on one side of the equation by adding $2$ to both sides

$| 3 x - 9 | - \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \le 7 + 2$

$| 3 x - 9 | \le 9$

Now, you have two possible cases to deal with

• $3 x - 9 \ge 0 \implies | 3 x - 9 | = 3 x - 9$

In this case, the inequality becomes

$3 x - 9 \le 9$

$3 x \le 18 \implies x \le \frac{18}{3} = 6$

• $3 x - 9 < 0 \implies | 3 x - 9 | = - \left(3 x - 9\right)$

In this case, the inequality becomes

$- \left(3 x - 9\right) \le 9$

$- 3 x + 9 \le 9$

$- 3 x \le 0 \implies x \ge 0$

You thus know that any value of $x$ that is smaller than or equal to $6$ and greater than or equal to $0$ will satisfy the original inequality. In interval notation, this is written as

$x \in \left(- \infty , 6\right] \cap \left[0 , + \infty\right)$

This means that the solution interval will be

$x \in \left[0 , 6\right]$