Which of the following expressions is equivalent to \frac{x^2 - 4x - 12}{2x - 12} - \frac{… — SAT

Which of the following expressions is equivalent to $\frac{x^2 - 4x - 12}{2x - 12} - \frac{x}{4}$ for any real number $x > 6$ ?

The expression stating that the equivalent form is the fraction with

x+4

in the numerator and

4

in the denominator

The correct equivalent expression is found by first factoring the rational expression

\frac{x^2 - 4x - 12}{2x - 12}

into

\frac{(x - 6)(x + 2)}{2(x - 6)}

. Since

x > 6

, the factor

x - 6

is non-zero and can be canceled, leaving

\frac{x + 2}{2}

. Converting

\frac{x + 2}{2}

to have a common denominator of

4

yields

\frac{2x + 4}{4}

. Finally, subtracting

\frac{x}{4}

gives the simplified result of

\frac{x + 4}{4}

Factor the quadratic numerator and the linear denominator of the first rational term.

\frac{x^2 - 4x - 12}{2x - 12} = \frac{(x - 6)(x + 2)}{2(x - 6)}

To identify and cancel common algebraic factors in the fraction.

Cancel the common factor

x - 6

from the numerator and denominator.

\frac{x + 2}{2}

Since

x > 6

, the term

x - 6

is non-zero and can be canceled.

Find a common denominator of

4

to perform subtraction with the second term,

-\frac{x}{4}

\frac{2(x + 2)}{4} - \frac{x}{4} = \frac{2x + 4}{4} - \frac{x}{4}

To subtract fractions, their denominators must be identical.

Subtract the numerators and combine terms over the common denominator.

\frac{2x + 4 - x}{4} = \frac{x + 4}{4}

Subtracting like terms in the numerator yields the final simplified expression.

Equivalent Algebraic Expressions

Instead of simplifying algebraically, you can substitute a value for

x

that is greater than

6

. For example, if

x = 8

, the original expression evaluates to

\frac{8^2 - 4(8) - 12}{2(8) - 12} - \frac{8}{4} = \frac{20}{4} - 2 = 3

. Evaluating the correct option with

x = 8

yields

\frac{8+4}{4} = 3

, confirming equivalence.

Estimated Time:1m 30s

Question