SOLVE AND GRAPH LINEAR ABSOLUTE VALUE INEQUALITIES
Absolute Value inequalities are translated into "and" or "or" compound statements before they
are solved. Therefore, the characteristics of the "and" / "or" statements learned in a previous
lesson (see class notes titled “solve & graph compound linear inequalities) still hold true for
both the algebraic expression and the graph.
PROPERTIES OF ABSOLUTE VALUE INEQUALITIES:
1) if |ax + b| < c, then (ax + b) < c and -(ax + b) < c
(if absolute value is less than/less than or equal to, then translates into "and" compound)
2) if |ax + b| > c, then (ax + b) > c or -(ax + b) > c
(if the absolute value is greater than/greater than or equal to, then translates into "or"
compound)
Guide to solving absolute value inequalities:
1)
2)
3)
4)
Isolate the absolute value, if necessary.
Separate into two equations joined by the correct compound statement.
Solve.
Graph.
EXAMPLES:
1) Solve and graph:
step 2:
step 3:
|2x - 5| < 13
2x - 5 < 13 and -(2x - 5) < 13
2x < 18
x<9
-2x + 5 < 13
-2x < 8
x > -4
The algebraic solution is x < 9 and x > -4.
step 4:
Graph (recall this is "and" compound, so solution is where graph overlaps/intersects)
x<9
9
<--------()----------------|-------------------------------------()------->
x > -4
-4
0
The final graph is <------()---------------------------------------------()--------->
-4
9
2) Solve and graph:
|9 + 3x| + 12 ≥ 48
step 1:
|9 + 3x| ≥ 36
step 2:
9 + 3x ≥ 36
step 3:
3x ≥ 27
x ≥ 9
(subtracted 12 on both sides)
or -(9 + 3x) ≥ 36
-9 - 3x ≥ 36
-3x ≥ 36
x ≤ -12
The algebraic solution is x ≥ 9 or x ≤ -12.
step 4: Graph (recall this is an "or" compound, solution is the union of both graphs)
<----------()------------------------|------------------------()--------------->
-12
0
9