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If |3a + 7| ≥ 2a + 12, then

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Bunuel
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If |3a + 7| ≥ 2a + 12, then [#permalink] If |3a + 7| ≥ 2a + 12, then   23 Jun 2021, 08:15
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A
B
C
D
E

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Question Stats:

79% (01:29) correct 21% (01:50) wrong based on 284 sessions
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If \(|3a + 7| ≥ 2a + 12\), then


(A) \(a ≤ -\frac{19}{5}\)

(B) \(a ≥ -\frac{19}{5}\)

(C) \(a ≥ 5\)

(D) \(a ≤ -\frac{19}{5}\) or \(a ≥ 5\)

(E) \(-\frac{19}{5} ≤ a ≤ 5\)
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D
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If |3a + 7| ≥ 2a + 12, then [#permalink] If |3a + 7| ≥ 2a + 12, then   23 Jun 2021, 08:33
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Bunuel wrote:
If \(|3a + 7| ≥ 2a + 12\), then


(A) \(a ≤ -\frac{19}{5}\)

(B) \(a ≥ -\frac{19}{5}\)

(C) \(a ≥ 5\)

(D) \(a ≤ -\frac{19}{5}\) or \(a ≥ 5\)

(E) \(-\frac{19}{5} ≤ a ≤ 5\)


Case-1: If a > 0


\(|3a + 7| ≥ 2a + 12\)
i.e. \(3a + 7 ≥ 2a + 12\)
i.e. \(3a -2a ≥ 12-7\)
i.e. \(a ≥ 5\)

Case-2: If -7/3 < a < 0


let, a = -x where x >0
\(|3a + 7| ≥ 2a + 12\)
i.e. \(-3x + 7 ≥ -2x + 12\)
i.e. \(x ≤ -5\)
i.e. \(a ≥ 5\)
NOT POSSIBLE i.e. Case-2 REJECTED

Case-3: If -7/3 > a


let, a = -x where x >0
\(|3a + 7| ≥ 2a + 12\)
i.e. \(3x - 7 ≥ -2x + 12\)
i.e. \(x ≥ 19/5\)
i.e. \(a ≤ -19/5\)

i.e. \(a ≤ -\frac{19}{5}\) or \(a ≥ 5\)



ANswer: Option D
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Re: If |3a + 7| ≥ 2a + 12, then [#permalink] If |3a + 7| ≥ 2a + 12, then   23 Jun 2021, 08:44
Asked: If \(|3a + 7| ≥ 2a + 12\), then

\(|3a + 7| ≥ 2a + 12\)

Case 1: a >= - 7/3
3a + 7 >= 2a + 12
a >=5
a>=5 > - 7/3

Case 2: a< -7/3
-3a - 7 >= 2a + 12
5a <= -19
a <= - 19/5 < - 7/3

IMO D
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If |3a + 7| 2a + 12, then [#permalink] If |3a + 7| ≥ 2a + 12, then   14 Jul 2023, 11:46
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Given that \(|3a + 7| ≥ 2a + 12\) and we need to find the range for a

\(|3a + 7| ≥ 2a + 12\)



We will have two cases
-Case 1: 3a + 7 ≥ 0

=> 3a ≥ -7
=> a ≥ \(\frac{-7}{3}\)
=> |3a + 7| = 3a + 7
=> 3a + 7 ≥ 2a + 12
=> a ≥ 5

And our condition was a ≥ \(\frac{-7}{3}\)
=> a ≥ 5 is a SOLUTION		-Case 2: 3a + 7 < 0

=> 3a < -7
=> a < \(\frac{-7}{3}\) ~ -2.3
=> |3a + 7| = -(3a + 7)
=> -3a - 7 ≥ 2a + 12 => 5a ≤ -19
=> a ≤ \(\frac{-19}{5}\) (=-3.8)

And our condition was a < \(\frac{-7}{3}\)
=> a ≤ \(\frac{-19}{5}\) is a SOLUTION


So, Answer will be D
Hope it helps!

Watch the following video to learn the Basics of Absolute Values

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If |3a + 7| 2a + 12, then [#permalink]
14 Jul 2023, 11:46
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Bunuel
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