Question

for which values of ( x ) is the function ( fleft(x
ight)=\frac{x^{2}+2x - 8}{x^{2}+7x - 8} ) negative?
( \bigcirc (-8, -4) ) and ( (1, 2) )
( \bigcirc (-4, -2) ) and ( (1, 8) )
( \bigcirc (-infty, -4) ) and ( (2, infty) )
( \bigcirc (-8, -4) ) and ( (-2, 1) )

Explanation:

Step1: Factor numerator and denominator

Numerator: $x^2+2x-8=(x+4)(x-2)$
Denominator: $x^2+7x-8=(x+8)(x-1)$
So $f(x)=\frac{(x+4)(x-2)}{(x+8)(x-1)}$

Step2: Find critical points

Set numerator/denominator to 0:
$x=-8, -4, 1, 2$
These split the number line into intervals:
$(-\infty,-8), (-8,-4), (-4,1), (1,2), (2,\infty)$

Step3: Test sign in each interval

• For $(-\infty,-8)$ (e.g., $x=-9$):

$\frac{(-)(-)}{(-)(-)}=\frac{+}{+}=+$ (positive)

• For $(-8,-4)$ (e.g., $x=-6$):

$\frac{(-)(-)}{(+)(-)}=\frac{+}{-}=-$ (negative)

• For $(-4,1)$ (e.g., $x=0$):

$\frac{(+)(-)}{(+)(-)}=\frac{-}{-}=+$ (positive)

• For $(1,2)$ (e.g., $x=1.5$):

$\frac{(+)(-)}{(+)(+)}=\frac{-}{+}=-$ (negative)

• For $(2,\infty)$ (e.g., $x=3$):

$\frac{(+)(+)}{(+)(+)}=\frac{+}{+}=+$ (positive)

Step4: Exclude undefined points

$x=-8,1$ make denominator 0, so they are not included.

Answer:

$(-8, -4)$ and $(1, 2)$