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If |-x/3+1|<2
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12 May 2025, 08:31
12 May 2025, 08:31
Expert Reply
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Question Stats:
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100% (03:10) wrong based on 1 sessions
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If $\(\left|\frac{-x}{3}+1\right|<2\)$, which of the following must be true?
(A) $\(x>0\)$
(B) $\(x<9\)$
(C) $\(x>-9\)$
(D) $\(0<x<3\)$
(A) $\(x>0\)$
(B) $\(x<9\)$
(C) $\(x>-9\)$
(D) $\(0<x<3\)$
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Re: If |-x/3+1|<2
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15 May 2025, 04:00
15 May 2025, 04:00
Expert Reply
Given Inequality:
$$
\(\left|\frac{-x}{3}+1\right|<2\)
$$
Step 1: Simplify the Inequality
$$
\(\left|-\frac{x}{3}+1\right|=\left|1-\frac{x}{3}\right|\)
$$
So, the inequality becomes:
$$
\(-2<1-\frac{x}{3}<2\)
$$
Step 2: Solve the Compound Inequality
1. Left Part:
$$
\(1-\frac{x}{3}>-2 \Longrightarrow-\frac{x}{3}>-3 \Longrightarrow x<9\)
$$
2. Right Part:
$$
\(1-\frac{x}{3}<2 \Longrightarrow-\frac{x}{3}<1 \Longrightarrow x>-3\)
$$
Final Solution:
$$
\(-3<x<9\)
$$
Step 3: Evaluate Each Option
- A: $\(x>0\)$
False. $x$ can be negative (e.g., $\(x=-1\)$ ).
- B: $\(x<9\)$
- True. Directly from the solution.
- $\(\mathbf{C :} x>-9\)$
- True. Since $\(x>-3\)$, and $\(-3>-9\)$, this is always true.
- D: $\(0<x<3\)$
False. $x$ can be outside this range (e.g., $x=4$ or $x=-1$ ).
Conclusion:
Options B and C must be true.
$$
\(\left|\frac{-x}{3}+1\right|<2\)
$$
Step 1: Simplify the Inequality
$$
\(\left|-\frac{x}{3}+1\right|=\left|1-\frac{x}{3}\right|\)
$$
So, the inequality becomes:
$$
\(-2<1-\frac{x}{3}<2\)
$$
Step 2: Solve the Compound Inequality
1. Left Part:
$$
\(1-\frac{x}{3}>-2 \Longrightarrow-\frac{x}{3}>-3 \Longrightarrow x<9\)
$$
2. Right Part:
$$
\(1-\frac{x}{3}<2 \Longrightarrow-\frac{x}{3}<1 \Longrightarrow x>-3\)
$$
Final Solution:
$$
\(-3<x<9\)
$$
Step 3: Evaluate Each Option
- A: $\(x>0\)$
False. $x$ can be negative (e.g., $\(x=-1\)$ ).
- B: $\(x<9\)$
- True. Directly from the solution.
- $\(\mathbf{C :} x>-9\)$
- True. Since $\(x>-3\)$, and $\(-3>-9\)$, this is always true.
- D: $\(0<x<3\)$
False. $x$ can be outside this range (e.g., $x=4$ or $x=-1$ ).
Conclusion:
Options B and C must be true.
gmatclubot
Re: If |-x/3+1|<2 [#permalink]
15 May 2025, 04:00
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