STRATEGY: As with all GRE Multiple Choice 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