MAML, Inequalities and Absolute Value: Different Cases
This is a MAML (Maine Association of Math Leagues) problem from October 2010 (Meet 1 of 2010-11), Round 2, Problem 2: http://maml.net/problems.htm
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Condensed Written Solution:
|4x-7|＞=|5x-6|
First, find when the absolute value expressions are negative and non-negative:
x ＜ 7/4 -＞ 4x-7 is negative; x ＞= 7/4 -＞ 4x-7 is non-negative
x ＜ 6/5 -＞ 5x-6 is negative; x ＞= 6/5 -＞ 5x-6 is non-negative
Now, 6/5 ＜ 7/4, so we have the following cases:
x ＜ 6/5 -＞ Both expressions are negative
6/5 ＜= x ＜ 7/4 -＞ 4x-7 is negative, 5x-6 is positive
x ＞= 7/4 -＞ Both expressions are positive
We can solve inequalities for each case:
x ＜ 6/5 -＞ 7-4x ＜ 6-5x -＞ x ＜= -1
x ＜= -1 ＜ 6/5 -＞ x ＜= -1
6/5 ＜= x ＜ 7/4 -＞ 7-4x ＜ 5x-6 -＞ x ＞= 13/9
6/5 ＜= 13/9 ＜= x ＜ 7/4 -＞ 13/9 ＜= x ＜ 7/4
x ＞= 7/4 -＞ 4x-7 ＜ 5x-6 -＞ x ＞= -1
-1 ＜ 7/4 ＜= x -＞ 7/4 ＜= x
Thus, we have the following solution:
x ＜= -1 OR 13/9 ＜= x ＜ 7/4 OR 7/4 ＜= x
Clearly, this simplifies to:
x ＜= -1 OR x ＞= 13/9 ANSWER