1973 AHSME Problems/Problem 22
Contents
Problem
The set of all real solutions of the inequality
is
Solution
We can do casework upon the value of . First, consider the case where both absolute values are positive, which is when
. In this case, the equation becomes
. This turns into
, which is a contradiction with our original assumption of
. Therefore, this case yields no solutions.
Next, consider the case when both absolute values are negative, which is when . This yields
, or
. Again, this yields a contradiction to the assumption that
, so this yields no solutions.
The next case is when the first absolute value is positive and the second is negative. This occurs when and
. Obviously, this also has no solutions.
The final case is when the first absolute value is negative and the second is positive. This occurs when and
. This yields
, which also has no solutions.
Therefore, there are no solutions and the answer is .
Note
The answer key lists the answer as , but graphing the expression, there are clearly no solutions. The answer given by the answer key would be correct if the equation were
As a result, either the answer key is incorrect or the problem is incorrect.
To see a graph of the expression, go here: https://www.desmos.com/calculator/iqj0rtqc92
See Also
1973 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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