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

1 Answer
Jun 19, 2017

Answer:

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:

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