Absolute Value Review
Math 208,
Fall 2009
In the following, let X (and Y) be any mathematical expression and p number. What appears below in bold must be memorized exactly or you will not be able to solve absolute value equations and inequalities.
- The equation | X | = p
is solved as follows:
- If p is positive, the solution is: X
= p OR X = –p
- If p is zero, the solution is: X
= 0
- If p is negative, there are no solutions
(i.e. contradiction).
- The equation | X | = |
Y | is solved by: X
= Y OR X = –Y
- The equation | X |
< p is solved by: –p < X < p
- The equation | X |
> p is solved by: X
> p OR X < –p
- Notes:
- Put equations into one of the above forms. E.g. make | t | + 6 < 10
into | t | < 4.
- Remember that double inequalities involve
AND. That is, these equations are
all equivalent:
–p < X < p –p < X AND X < p X > –p AND X < p.
It is probably easiest to remember the first of these for | X | < p. - The rule for < also works for ≤.
Similarly, the rule for > also works for ≥.
- In all the inequalities | X | < p,
| X | ≥ p, etc., the answer will be rather strange if
p is negative or zero. The
"usual" case is when p is positive.
Revised Tuesday, July 21, 2009. E-mail corrections, suggestions to mmaltenfort@ccc.edu