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

1 Answer
Douglas K.
Nov 1, 2017

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#
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