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If 1<|a|<3 and 5>b>2, then which of the following inequalities could b
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01 Sep 2025, 01:19
01 Sep 2025, 01:19
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If $\(1<|a|<3\)$ and $\(5>b>2\)$, then which of the following inequalities could be true?
Indicate all such inequalities.
(A) $\(-4<a b<0\)$
(B) $\(-2<a b<2\)$
(C) $\(2<a b<15\)$
(D) $\(5<a b<6\)$
Indicate all such inequalities.
(A) $\(-4<a b<0\)$
(B) $\(-2<a b<2\)$
(C) $\(2<a b<15\)$
(D) $\(5<a b<6\)$
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Re: If 1<|a|<3 and 5>b>2, then which of the following inequalities could b
[#permalink]
04 Oct 2025, 12:38
04 Oct 2025, 12:38
Expert Reply
Let's analyze the possible values for $a b$ given $\(1<|a|<3\)$ and $\(5>b>2\)$.
- If $\(a>1\)$, then $\(1<a<3\)$
- If $\(a<-1\)$, then $\(-3<a<-1\)$
- For all $\(b: 2<b<5\)$
Possible Ranges for $a b$
Case 1: $\(a>1\)$
- $\(a b>2 \times 1=2\)$ (lowest possible)
- $\(a b<3 \times 5=15\)$ (highest possible)
- So for $\(a>1: 2<a b<15\)$
Case 2: $a<-1$
- $\(a b>(-3) \times 5=-15\)$
- $\(a b<(-1) \times 2=-2\)$
- So for $\(a<-1:-15<a b<-2\)$
Check each option:
(A) $\(-4<a b<0\)$ :
Possible for $\(a<-1\)$ and $\(b>2\)$, since $\(-15<a b<-2\)$ covers part of $\(-4<a b<0\)$.
Correct.
(B) $\(-2<a b<2\)$ :
For $\(a>1, a b>2\)$, and for $\(a<-1, a b<-2 . a b\)$ cannot be between -2 and 2 under these conditions.
Not possible.
(C) $\(2<a b<15\)$ :
For $\(a>1,2<a b<15\)$. This is fully within the possible range.
Correct.
(D) $\(5<a b<6\)$ :
For $\(a>1\)$, possible. For example, $\(a=2, b=3\)$ gives $\(a b=6\)$, so yes, this interval is possible.
Correct.
Final answer:
The inequalities that could be true are:
$\(\mathrm{A}, \mathrm{C}\)$, and D .
- If $\(a>1\)$, then $\(1<a<3\)$
- If $\(a<-1\)$, then $\(-3<a<-1\)$
- For all $\(b: 2<b<5\)$
Possible Ranges for $a b$
Case 1: $\(a>1\)$
- $\(a b>2 \times 1=2\)$ (lowest possible)
- $\(a b<3 \times 5=15\)$ (highest possible)
- So for $\(a>1: 2<a b<15\)$
Case 2: $a<-1$
- $\(a b>(-3) \times 5=-15\)$
- $\(a b<(-1) \times 2=-2\)$
- So for $\(a<-1:-15<a b<-2\)$
Check each option:
(A) $\(-4<a b<0\)$ :
Possible for $\(a<-1\)$ and $\(b>2\)$, since $\(-15<a b<-2\)$ covers part of $\(-4<a b<0\)$.
Correct.
(B) $\(-2<a b<2\)$ :
For $\(a>1, a b>2\)$, and for $\(a<-1, a b<-2 . a b\)$ cannot be between -2 and 2 under these conditions.
Not possible.
(C) $\(2<a b<15\)$ :
For $\(a>1,2<a b<15\)$. This is fully within the possible range.
Correct.
(D) $\(5<a b<6\)$ :
For $\(a>1\)$, possible. For example, $\(a=2, b=3\)$ gives $\(a b=6\)$, so yes, this interval is possible.
Correct.
Final answer:
The inequalities that could be true are:
$\(\mathrm{A}, \mathrm{C}\)$, and D .
gmatclubot
Re: If 1<|a|<3 and 5>b>2, then which of the following inequalities could b [#permalink]
04 Oct 2025, 12:38
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