Question

How do you graph and solve `|4x + 1| >= 5`?

Answer 1

Use the definition of the absolute value function:

`|A| = {(A;A>=0),(-A;A <0):}`

to write two inequalities.

Simplify the restrictions.

Solve both inequalities.

Graph.

Explanation:

Given: `|4x+1|>=5`

Use the definition of the absolute value function to write two inequalities:

`4x+1 >=5;4x+1>=0` and `-4x-1>=5;4x+1<0`

Simplify the restrictions.

`4x+1 >=5;4x>=-1` and `-4x-1>=5;4x<-1`

`4x+1 >=5;x>=-1/4` and `-4x-1>=5;x<-1/4`

Solve the inequalities:

`4x >=4;x>=-1/4` and `-4x>=6;x<-1/4`

`x >=1;x>=-1/4` and `x<=-3/2;x<-1/4`

Because the inequalities do not violate the restrictions, we can drop them:

`x >=1` and `x<=-3/2`

To graph the first inequality you draw a solid vertical line at `x = 1` and shade in the area to the right of that line.

On the same graph, represent the second inequality by drawing a solid vertical line at `x = -3/2` and then shade in the area to the right of that line.

Here is that graph: