If x satisfies the equation \frac{5(x - 2)}{3} - \frac{2x - 1}{4} = \frac{x + 7}{6}, which… — GRE General Test

If $x$ satisfies the equation $\frac{5(x - 2)}{3} - \frac{2x - 1}{4} = \frac{x + 7}{6}$ , which of the following statements must be true? Select all that apply.

$x > 4$ Answer
$4x$ is an integerAnswer
C
$x$ is an integer
D
$2x + 1 > 10$
$3x - 2 = 10.75$ Answer

Answer

The correct statements are that

x > 4

4x

is an integer, and

3x - 2 = 10.75

Solving the linear equation yields

x = 4.25

\frac{17}{4}

. Using this value:

4.25 > 4

is true;

4 \times \frac{17}{4} = 17

is an integer, so that statement is true; and

3(4.25) - 2 = 10.75

is also true.

Step-by-Step Solution

Find a common denominator to clear fractions in the given equation.

The least common multiple of denominators

3

4

, and

6

12

Clearing denominators simplifies multi-term fractional equations into standard linear form.

Multiply every term of the equation by

12

and expand numerators.

4 \cdot 5(x - 2) - 3 \cdot (2x - 1) = 2 \cdot (x + 7) \implies 20(x - 2) - 3(2x - 1) = 2(x + 7)

Multiplying each term by

12

eliminates all fraction bars.

Distribute the constants through the parentheses on both sides.

20x - 40 - 6x + 3 = 2x + 14

Applying the distributive property correctly accounts for negative signs across parentheses.

Combine like terms on the left side and solve for

x

14x - 37 = 2x + 14 \implies 12x = 51 \implies x = \frac{51}{12} = \frac{17}{4} = 4.25

Isolating

x

gives the exact rational value of the solution.

Evaluate each provided statement using

x = 4.25

The statement

x > 4

is true (

4.25 > 4

). The statement

4x

is an integer is true (

4 \times 4.25 = 17

). The statement

3x - 2 = 10.75

is true (

3 \times 4.25 - 2 = 10.75

). The other two statements are false.

Direct substitution verifies which conditions hold.

Key Concept

Solving multi-step linear equations in one variable by clearing denominators and combining variable terms.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

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