1973 AHSME Problems/Problem 22
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 . Combining this with our original assumption, we get the solutions .
Next, consider the case when both absolute values are negative, which is when . This yields , or . Combining this with our original assumption, we get
The next case is when the first absolute value is positive and the second is negative. This occurs when and . Obviously, this has no solutions, since the inequalities do not overlap.
The final case is when the first absolute value is negative and the second is positive. This occurs when and . This yields , which is always true. Therefore, we also get the solutions .
Therefore, after combining all of our solutions, we get the range , which is .
See Also
1973 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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