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

Jun 19, 2017

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: $| 4 x + 1 | \ge 5$

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:

$x \ge 1$ and $x \le - \frac{3}{2}$

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 $x = - \frac{3}{2}$ and then shade in the area to the right of that line.

Here is that graph: 