If x is the solution to the linear equation \frac{4x - 3}{3} - \frac{x + 2}{4} = \frac{5x}… — GRE General Test

If $x$ is the solution to the linear equation $\frac{4x - 3}{3} - \frac{x + 2}{4} = \frac{5x}{6}$ , which of the following statements about $x$ must be true? Select all such statements.

$x$ is a multiple of 3Answer
$2x - 5 > 6$ Answer
$x^2 - 5x = 6$ Answer
D
$x$ is a prime number
E
$x < 5$

Answer

The statements establishing that

x

is a multiple of 3, that

2x - 5 > 6

, and that

x^2 - 5x = 6

are all true.

Solving the linear equation yields

x = 6

. Substituting

x = 6

into each statement shows that

x

is a multiple of 3 (

6 = 3 \times 2

2x - 5 > 6

evaluates to

7 > 6

, and

x^2 - 5x = 6

evaluates to

36 - 30 = 6

. All three statements are correct.

Step-by-Step Solution

Clear denominators by multiplying both sides by the least common multiple.

12 \cdot \left(\frac{4x - 3}{3} - \frac{x + 2}{4}\right) = 12 \cdot \left(\frac{5x}{6}\right) \implies 4(4x - 3) - 3(x + 2) = 2(5x)

The least common multiple of 3, 4, and 6 is 12.

Distribute factors and combine like terms.

16x - 12 - 3x - 6 = 10x \implies 13x - 18 = 10x

Distributing

-3

into

(x + 2)

yields

-3x - 6

Isolate

x

on one side of the equation.

13x - 10x = 18 \implies 3x = 18 \implies x = 6

Subtract

10x

and add 18 to isolate the variable term.

Evaluate each given statement using

x = 6

Statement 1:

6

is a multiple of 3 (True). Statement 2:

2(6) - 5 = 7 > 6

(True). Statement 3:

6^2 - 5(6) = 6

(True). Statement 4: 6 is prime (False). Statement 5:

6 < 5

(False).

Test each option against the calculated solution

x = 6

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

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