# How do you graph and solve 4<|3x-10|<15?

Nov 1, 2017

Use |3x-10|={(3x-10;x>=10/3),(-3x+10;x<10/3):} to separate the inequality into two inequalities.
Solve the inequalities and resolve any domain conflicts.
Graph on a number line.

#### Explanation:

Separate into two inequalities:

4<-3x+10<15; x < 10/3" [1]"
4<3x-10<15;x>=10/3" [2]"

Subtract 10 from all in inequality [1] and add 10 to all in inequality [2]:

-6<-3x<5; x < 10/3" [1.1]"
14<3x<25;x>=10/3" [2.1]"

Divide inequality [1.1] by -3 and divide inequality [2.1] by 3:

2>x>(-5)/3; x < 10/3" [1.2]"
14/3 < x < 25/3;x>=10/3" [2.2]"

There are no domain violations, therefore, we can drop the restrictions and write inequality [1.2] in a better form:

$\frac{- 5}{3} < x < 2 \text{ [1.3]}$
$\frac{14}{3} < x < \frac{25}{3} \text{ [2.2]}$

Graphing instructions:

1. Place circles on a number line at, $\frac{- 5}{3} , 2 , \frac{14}{3} , \mathmr{and} \frac{25}{3}$
2. Shade in the area between $\frac{- 5}{3} \mathmr{and} 2 ,$
3. Shade in the area between $\frac{14}{3} \mathmr{and} \frac{25}{3}$