To determine for which values of [tex]\( x \)[/tex] the rational expression
[tex]\[
\frac{x-7}{2x^2 - 32}
\][/tex]
is undefined, we need to identify the values of [tex]\( x \)[/tex] that make the denominator equal to zero.
Start by setting the denominator equal to zero and solving for [tex]\( x \)[/tex]:
[tex]\[
2x^2 - 32 = 0
\][/tex]
First, add 32 to both sides of the equation:
[tex]\[
2x^2 = 32
\][/tex]
Next, divide both sides of the equation by 2:
[tex]\[
x^2 = 16
\][/tex]
To solve for [tex]\( x \)[/tex], take the square root of both sides:
[tex]\[
x = \pm 4
\][/tex]
This means [tex]\( x = 4 \)[/tex] and [tex]\( x = -4 \)[/tex] are the values for which the denominator is zero and thus, the rational expression is undefined.
Among the given options:
- A. [tex]\(-2\)[/tex]
- B. [tex]\(7\)[/tex]
- C. [tex]\(4\)[/tex]
- D. [tex]\(-4\)[/tex]
- E. [tex]\(-7\)[/tex]
- F. [tex]\(2\)[/tex]
The correct values that make the rational expression undefined are [tex]\( x = 4 \)[/tex] and [tex]\( x = -4 \)[/tex].
Therefore, the correct answers are:
- C. [tex]\(4\)[/tex]
- D. [tex]\(-4\)[/tex]
