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07 Mar 2022, 08:41
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Difficulty:
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Question Stats:
72% (02:07) correct
28%
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wrong
based on 585
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If \(|3 - \frac{3x}{2}| ≥ 1\), which of the following is NOT a possible value of \(x\)?
A) \(-\frac{4}{3}\)
B) \(-\frac{1}{3}\)
C) \(\frac{4}{3}\)
D) \(\frac{5}{3}\)
E) \(\frac{8}{3}\)
A) \(-\frac{4}{3}\)
B) \(-\frac{1}{3}\)
C) \(\frac{4}{3}\)
D) \(\frac{5}{3}\)
E) \(\frac{8}{3}\)
Solving this question helps.
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08 Mar 2022, 02:36
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As we are dealing with an absolute value, we can write \(|3 - \frac{3x}{2|}≥1\) in the following two ways:
1) \(3 - \frac{3x}{2} ≥ 1\) and 2) \(3 - \frac{3x}{2} ≤ -1\)
1) \(3 - \frac{3x}{2} ≥ 1\)
multiply through by 2
\(6 - 3x ≥ 2\)
\(-3x ≥ -4 \)
Multiply through by -1
\(x ≤ \frac{4}{3}\)
2) \(3 - \frac{3x}{2} ≤ -1\)
multiply through by 2
\(6 - 3x ≤ -2\)
\(-3x ≤ -8\)
Multiply through by -1
\(3x ≥ 8\)
\(x ≥ \frac{8}{3}\)
So we know that \(x ≤ \frac{4}{3}\) or \(x ≥ \frac{8}{3}\). Which means that the numbers which do not satisfy the inequality are: \(\frac{5}{3} \);\(\frac{6}{3} \) & \( \frac{7}{3}\).
Answer choice D is the only choice which is cannot be a value of x.
Answer D
1) \(3 - \frac{3x}{2} ≥ 1\) and 2) \(3 - \frac{3x}{2} ≤ -1\)
1) \(3 - \frac{3x}{2} ≥ 1\)
multiply through by 2
\(6 - 3x ≥ 2\)
\(-3x ≥ -4 \)
Multiply through by -1
\(x ≤ \frac{4}{3}\)
2) \(3 - \frac{3x}{2} ≤ -1\)
multiply through by 2
\(6 - 3x ≤ -2\)
\(-3x ≤ -8\)
Multiply through by -1
\(3x ≥ 8\)
\(x ≥ \frac{8}{3}\)
So we know that \(x ≤ \frac{4}{3}\) or \(x ≥ \frac{8}{3}\). Which means that the numbers which do not satisfy the inequality are: \(\frac{5}{3} \);\(\frac{6}{3} \) & \( \frac{7}{3}\).
Answer choice D is the only choice which is cannot be a value of x.
Answer D
08 Mar 2022, 08:57
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BrentGMATPrepNow
STRATEGY: As with all GMAT Problem Solving questions, we should immediately ask ourselves, Can I use the answer choices to my advantage?
In this case, we could test the answer choices, but doing so would involve evaluating fractions within fractions, which would be quite time-consuming.
Now we should give ourselves about 10-20 seconds to identify a faster approach.
In this case, it will likely be faster to use algebra to solve the given inequality for x. So let's do that.
Two properties involving absolute value inequalities:
Property #1: If |something| < k, then –k < something < k
Property #2: If |something| > k, then EITHER something > k OR something < -k
Note: these rules assume that k is positive
The given inequality is in the form of Property #2, which means...
Either \(3 - \frac{3x}{2} ≥ 1\) or \(3 - \frac{3x}{2} ≤ -1\).
Let's solve each inequality for \(x\)
Take: \(3 - \frac{3x}{2} ≥ 1\)
Subtract \(3\) from both sides: \(-\frac{3x}{2} ≥ -2\)
Multiply both sides by \(2\) to get: \(-3x ≥ -4\)
Divide both sides by \(-3\) to get: \(x ≤ \frac{4}{3}\) [ since we divided both sides of the inequality by a negative value, we reversed the direction of the inequality symbol]
Take: \(3 - \frac{3x}{2} ≤ -1\)
Subtract \(3\) from both sides: \(-\frac{3x}{2} ≤ -4\)
Multiply both sides by \(2\) to get: \(-3x ≤ -8\)
Divide both sides by \(-3\) to get: \(x ≥ \frac{8}{3}\)
If \(x ≤ \frac{4}{3}\) or \(x ≥ \frac{8}{3}\), then \(x\) cannot equal \(\frac{5}{3}\)
Answer: D
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