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# If |3 - 3x/2| 1, which of the following is NOT a possible value of

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If |3 - 3x/2| 1, which of the following is NOT a possible value of [#permalink]
Properties of Modulus

Property 1: the output of the negation of the quantity inside the absolute value will be the SAME

In other words: | X - k | = | k - X |

Property 2:

| kX - kA | ——> you can extract a common factor and multiply absolute values

Assuming k = some constant number value

| kX - kA | = | k | * |X - A| = k * |X - A|

Starting with the question stem:

|3 - 3x/2 | = | 3x/2 - 3|

So we have

|3x/2 - 3| >/= 1

Multiply both sides of the inequality by the absolute value of |2| and use property 2 above:

|2| * |3x/2 - 3| >/= 1 * |2|

|3x - 6| >/= |2|

Use property 2 again to extract common fact on left side:

3 * |x - 2| >/= 2

|x - 2| >/= 2/3

X is at a distance of greater than or equal to 2/3rd units away from +2 on the number line

This means:

X </= 4/3

Or

X >/= 8/3

Only 5/3 does not fall in the solution ranges of X

5/3 is in the middle part between 4/3 and 8/3 that does not satisfy the inequality

*D*

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Re: If |3 - 3x/2| 1, which of the following is NOT a possible value of [#permalink]
BrentGMATPrepNow wrote:
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 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 $$\frac{5}{3}$$

RELATED VIDEO

Hi BrentGMATPrepNow, is this due to ≤ sign in 8/3 ≤ x ≤ 4/3, Therefore we are looking for an equivalent value of 8/3 & 4/3 rather than a between value such as 5/3? Thanks Brent
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Re: If |3 - 3x/2| 1, which of the following is NOT a possible value of [#permalink]
Top Contributor
Kimberly77 wrote:

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 $$\frac{5}{3}$$

RELATED VIDEO

Hi BrentGMATPrepNow, is this due to ≤ sign in 8/3 ≤ x ≤ 4/3, Therefore we are looking for an equivalent value of 8/3 & 4/3 rather than a between value such as 5/3? Thanks Brent[/quote]

Be careful.
If we know that $$x ≤ \frac{4}{3}$$ or $$x ≥ \frac{8}{3}$$, we can't combine the two inequalities to get 8/3 ≤ x ≤ 4/3

Notice that if we remove the x, your inequality becomes 8/3 ≤ 4/3, which isn't true.
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Re: If |3 - 3x/2| 1, which of the following is NOT a possible value of [#permalink]
Noted thanks BrentGMATPrepNow for the clarification. So we are looking for what x is not equal to here rather than between value? Make sense now thanks Brent
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Re: If |3 - 3x/2| 1, which of the following is NOT a possible value of [#permalink]
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Re: If |3 - 3x/2| 1, which of the following is NOT a possible value of [#permalink]
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