To determine the values of [tex]\( x \)[/tex] for which the rational expression [tex]\( \frac{x-7}{2x^2 - 32} \)[/tex] is undefined, we need to identify where the denominator is zero. The rational expression is undefined whenever the denominator equals zero because division by zero is undefined in mathematics.
The denominator of the rational expression is [tex]\( 2x^2 - 32 \)[/tex]. We need to solve the equation:
[tex]\[ 2x^2 - 32 = 0 \][/tex]
First, we isolate the term involving [tex]\( x^2 \)[/tex]:
[tex]\[ 2x^2 - 32 = 0 \][/tex]
[tex]\[ 2x^2 = 32 \][/tex]
Next, we divide both sides by 2 to simplify:
[tex]\[ x^2 = 16 \][/tex]
To find [tex]\( x \)[/tex], we take the square root of both sides. Remember, taking the square root of a number gives both a positive and a negative solution:
[tex]\[ x = \pm \sqrt{16} \][/tex]
[tex]\[ x = \pm 4 \][/tex]
Thus, the values of [tex]\( x \)[/tex] that make the denominator equal to zero (and hence the rational expression undefined) are:
[tex]\[ x = 4 \][/tex]
[tex]\[ x = -4 \][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which the rational expression [tex]\( \frac{x-7}{2x^2 - 32} \)[/tex] is undefined are:
E. 4
F. -4
