If \left|- x3+1\right|<2, which of the following must be true?

Free Practice Question

PS · Problem Solving

705-805

Inequalities

If $\left|-\frac{x}{3}+1\right|<2$ , which of the following must be true?

Answer Choices

Correct answer marked below

x>0

x<8

x>-4

Correct

0<x<3

None of the above

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Comments

SaketAug 16

This one got me. I initially just multiplied both sides by 3 and lost the sign flip on the negative, ended up with x < 9 and no matching choice. Took me a minute to see I had to split into two cases.

ColinAug 22

Same trap here. The minus sign in front of x/3 is brutal if you rush.

SohiniAug 26

I did the exact same thing and stared at the answer choices way too long lol.

VishalMay 6

|−x/3 + 1| < 2 means −2 < −x/3 + 1 < 2. From there subtract 1 and carefully flip when multiplying by −3. Not bad once you set it up, but a 700+ level question for sure.

AshwantMay 16

Yep, subtracting 1 first then handling the −3 is way cleaner than splitting cases.

ShobhitAug 1

Is anyone else bothered that one of the answer choices is a strict two-sided interval while the others are one-sided? I always feel like that's a hint the trap is about what 'must be true' actually means vs what's 'equivalent.'

ArushiAug 13

Totally. 'Must be true' vs 'the solution set' is a different question. Easy to pick the exact range and forget it has to cover every case.

MehakJun 15

Got the range pretty fast, but then the answer choices don't have it and I'm second-guessing. Is the point of this one that the full solution isn't listed, so only a weaker statement must hold?

JosephJun 25

That's exactly where I landed. The solution set is more specific than any single option, so you have to test which one-sided statement is always implied.

What this question tests

What it tests

Your ability to solve and combine linear inequalities and reason about ranges of values.

Common trap

Failing to flip the inequality sign when multiplying or dividing by a negative, or missing an inclusive/exclusive endpoint.

If ∣−x3+1∣<2\left|-\frac{x}{3}+1\right|<2​−3x​+1​<2, which of the following must be true?