BrentGMATPrepNow wrote:
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) \(\frac{8}{3}\)
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 positiveThe 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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