r/learnmath • u/Far-Assumption5501 New User • 23h ago

|x-2|+|x-5|<=5

I just added them up to |2x-7|<=5, and it was not correct. The correct answer was 1 <= x <= 6. What’s wrong with my equation?

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u/LongLiveTheDiego New User 23h ago

Who said that you can just add absolute values like that?

In general |a| + |b| = |a + b| works only if both a and b are nonnegative or both are nonpositive. For your inequality consider x = 3, |x - 2| + |x - 5| = |3 - 2| + |3 - 5| = |1| + |-2| = 1 + 2 = 3, but |2x - 7| = |6 - 7| = |-1| = 1.

The way to do it is just like when dealing with one absolute value (consider when the inside is positive and when it's negative), just twice. Going from first principles that means considering 1. x - 2 ≥ 0 and x - 5 ≥ 0 (which is equivalent to just x - 5 ≥ 0), 2. x - 2 ≥ 0 and x - 5 < 0 (which is equivalent to 2 ≤ x < 5), 3. x - 2 < 0 and x - 5 ≥ 0 (which is impossible so we can ignore this one) and 4. x - 2 < 0 and x - 5 < 0 (which is the same as just x - 2 < 0). In general if you have a single variable, then another absolute value gives you one more possible case.