Question

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

Answer 1

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:

`(-5)/3 < x < 2" [1.3]"`
`14/3 < x < 25/3" [2.2]"`

Graphing instructions:

1. Place circles on a number line at, `(-5)/3, 2, 14/3, and 25/3`
2. Shade in the area between `(-5)/3 and 2,`
3. Shade in the area between `14/3 and 25/3`