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Linear Absolute Value Inequality: Solving Algebraically
Example 1: Solve 4 3x  2  12  0.
First, isolate the absolute value
bar expression.
4 3x  2  12
3x  2  3
Next, examine the constant on the right side.
If it is positive or zero proceed as follows. Write two new
inequalities. Form one by dropping the absolute value
bars, changing the sign of the constant term and changing
the direction of the inequality symbol. Form the other by
dropping the absolute value bars only. Place the word "or"
between them.
3x  2   3 or 3x  2  3
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Linear Absolute Value Inequality: Solving Algebraically
Now solve each inequality. 3x  2   3 or 3x  2  3
The "or" means that any
3x   1 or 3x  5
number that satisfies either
1
5
x   or x 
of the inequalities will be a
3
3
solution of the original
inequality.
1
5 


The solution set in interval notation is   ,     ,  .
3

3 
Notes: If after isolating the absolute value expression the
constant term on the right is negative, the solution set of the
inequality would be (- , ) because any real number
substituted for x will cause the absolute value expression to
produce a number greater than a negative number.
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Slide 2
Linear Absolute Value Inequality: Solving Algebraically
Example 2: Solve 4 3x  2  12  0.
First, isolate the absolute value
bar expression.
4 3x  2  12
3x  2  3
Next, examine the constant on the right side.
If it is positive or zero proceed as follows. Write two new
inequalities. Form one by dropping the absolute value
bars, changing the sign of the constant term and changing
the direction of the inequality symbol. Form the other by
dropping the absolute value bars only. Place the word
"and" between them.
3x  2   3 and 3x  2  3
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Slide 3
Linear Absolute Value Inequality: Solving Algebraically
Now solve each inequality. 3x  2   3 and 3x  2  3
The "and" means that only
3x   1 and 3x  5
those numbers that satisfy
1
5
x   and x 
both of the inequalities will
3
3
be solutions of the original
  1 , 5 .
inequality. The solution set in interval notation is 

 3 3
Notes: If after isolating the absolute value expression the
constant term on the right is negative, the inequality would
have no solutions because no real number substituted for x
will cause the absolute value expression to produce a
number less than a negative number.
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Slide 4
Linear Absolute Value Inequality: Solving Algebraically
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