Question

How do you solve `|9x + 1| < - 4`?

Answer 1

There is no solution.

Explanation:

Given:

`|9x + 1| < - 4`

Use the piecewise definition of the absolute value function

`|9x+1|={(9x+1; x >=-1/9),(-(9x+1);x < -1/9):}`

and separate the inequality into two inequalities:

`9x + 1 < - 4; x >= -1/9` and `-(9x+1)<-4; x < -1/9`

Multiply the second inequality by -1:

`9x + 1 < - 4; x >= -1/9` and `9x+1 > 4; x < -1/9`

Subtract 1 from both sides of both inequalities:

`9x < - 5; x >= -1/9` and `9x > 3; x < -1/9`

Divide both sides of both inequalities by 9:

`x < - 5/9; x >= -1/9` and `x > 1/3; x < -1/9`

Please observe that x cannot be less than `-5/9` and greater than or equal to `-1/9` and x cannot be greater than `1/3` and less than `-1/9`. Therefore, there is no solution.