# How do you graph and solve  |3-2x|>7?

Jan 12, 2018

Solution: color(blue)(x<-2 or x >5

Using Interval Notation: $\textcolor{b l u e}{\left(- \infty , - 2\right) \cup \left(5 , \infty\right)}$

#### Explanation:

Given the Inequity Expression:

$\textcolor{red}{| 3 - 2 x | > 7}$

We must remember that,

$\textcolor{b l u e}{| f \left(x\right) | > a \Rightarrow f \left(x\right) < - a \mathmr{and} f \left(x\right) > a}$

Hence,

we consider two possibilities as shown below:

$\textcolor{g r e e n}{3 - 2 x < - 7 \mathmr{and} 3 - 2 x > 7}$

First, we will consider

color(red)(3-2x < -7

Subtract 3 from both sides

$\Rightarrow 3 - 2 x - 3 < - 7 - 3$

$\Rightarrow \cancel{3} - 2 x - \cancel{3} < - 7 - 3$

Simplify to get

$\Rightarrow - 2 x < - 10$

Multiply both the sides by $\left(- 1\right)$

We must remember to reverse the inequality

$\Rightarrow - 2 x \left(- 1\right) > - 10 \left(- 1\right)$

Simplify to get

$\Rightarrow 2 x > 10$

Divide both sides by the value of $\left(2\right)$

$\Rightarrow \frac{2 x}{2} > \frac{10}{2}$

$\Rightarrow \frac{\cancel{2} x}{\cancel{2}} > \frac{10}{2}$

$\Rightarrow x > 5$ Intermediate Result 1

Next, we will consider

color(red)(3-2x >7

Subtract the value of 3 from both sides

$3 - 2 x - 3 > 7 - 3$

$\Rightarrow \cancel{3} - 2 x - \cancel{3} > 7 - 3$

$\Rightarrow - 2 x > 4$

Multiply both the sides by $\left(- 1\right)$

We must remember to reverse the inequality

$\Rightarrow - 2 x \left(- 1\right) < 4 \left(- 1\right)$

$\Rightarrow 2 x < - 4$

Divide both sides by the value of 2

$\Rightarrow \frac{2 x}{2} < - \frac{4}{2}$

$\Rightarrow \frac{\cancel{2} x}{\cancel{2}} < - \frac{4}{2}$

$\Rightarrow x < - 2$ Intermediate Result 2

Let us now combine both the intermediate results (1) & (2) to get

$\textcolor{b l u e}{x < - 2 \mathmr{and} x > 5}$

Using the Interval Notation we can can also write the solution as

$\textcolor{g r e e n}{\left(- \infty , - 2\right) \cup \left(5 , \infty\right)}$

Please refer to the inequality graph below which demonstrates our solution: