What is the value of x that satisfies the equation \frac{3}{4}(x - 8) = \frac{1}{2}x + 2? — ACT

What is the value of $x$ that satisfies the equation $\frac{3}{4}(x - 8) = \frac{1}{2}x + 2$ ?

Answer: 32

The correct value of

x

32

Distributing

\frac{3}{4}

on the left side yields

\frac{3}{4}x - 6

. Subtracting

\frac{1}{2}x

(which is

\frac{2}{4}x

) from both sides gives

\frac{1}{4}x - 6 = 2

. Adding 6 to both sides gives

\frac{1}{4}x = 8

, and multiplying by 4 yields the correct value of

32

Distribute the fraction

\frac{3}{4}

to both terms inside the parentheses.

\frac{3}{4}x - 6 = \frac{1}{2}x + 2

To simplify the equation by removing the parentheses.

Subtract

\frac{1}{2}x

from both sides of the equation.

\frac{1}{4}x - 6 = 2

To collect all terms with the variable

x

on one side of the equation.

Add 6 to both sides of the equation.

\frac{1}{4}x = 8

To isolate the term containing the variable.

Multiply both sides of the equation by 4.

x = 32

To solve for

x

by eliminating the coefficient of

\frac{1}{4}

Solving linear equations with variables on both sides and fractional coefficients

Multiply the entire equation by the least common denominator of the fractions, which is 4, to clear the fractions before solving:

4 \cdot [\frac{3}{4}(x - 8)] = 4 \cdot [\frac{1}{2}x + 2] \implies 3(x - 8) = 2x + 8

. Then distribute:

3x - 24 = 2x + 8

. Subtract

2x

from both sides:

x - 24 = 8

. Add 24 to both sides:

x = 32

Estimated Time:45s

Question