MATH 0310 / MATH 1314
Absolute Value Equations & Inequalities
1. |x| = n
are those numbers that satisfy x = n “or” x = -n
2. |x| = -n
D.N.E. (does not exist),
3. |x| < n
are those numbers that satisfy x > -n “and” x < n
4. |x| > n
are those numbers that satisfy x < -n “or” x > n
5. |x| > -n
satisfies all Real Numbers (R)
6. |x| < -n
D.N.E. (does not exist),
(null set)
(null set)
Graphs: Absolute Value graphs are in the shape of the letter V.
y = |x|
y = a|x + b| + c
The value of “a” causes the “V” to point and fold up or down:
if “a” > 0 (positive), the “V” points up.
if “a” < 0 (negative), the “V” points down.
if “a” > 1, “V” graph folds up.
if “a” < 1, “V” graph folds down.
“b” moves the “V” horizontally opposite to the sign of “b”.
“c” moves the “V” vertically according to the sign of “c”.
Examples of y = a|x+b| + c:
y = |x|
a= +1, b=0, c=0
Figure 1.
y = 2|x|
a= +2, b=0, c=0
Figure 3.
y = 2|x+1|+2
a = +1, b= +1, c= +2
Figure 5.
LSC-Montgomery Learning Center: Absolute Value Equations and Inequalities
Last Updated April 13, 2011
y = -|x|
a= -1, b=0, c=0
Figure 2.
y = 2|x+1|
a= +2, b= +1, c=0
Figure 4.
y=
1
|x+1|+2
2
a= + 12 , b= +1, c= +2
Figure 6.
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