We are given the following function and asked to find its

x -

intercepts.

f (x) = \frac{2}{x - 1} - \frac{x + 4}{3}

Let's start with recalling that an

x -

intercept is a point where the graph intersects the

x -

axis. The

y -

coordinate of this point is

0 .

Hence, to find the

x -

intercepts of the given function we can substitute

f (x)

with

0

and solve it for

x .

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

First, we will rewrite the right-hand side as one rational expression. In order to do this, we will expand the first fraction by

3

and the second one by

x - 1

so that they have a common denominator.

0 = \frac{2 ( 3 )}{3 ( x - 1 )} - \frac{( x + 4 ) ( x - 1 )}{3 ( x - 1 )}

Now we are able to subtract the fractions.

0 = \frac{2 ( 3 )}{3 ( x - 1 )} - \frac{( x + 4 ) ( x - 1 )}{3 ( x - 1 )}

▼

Simplify right-hand side

Subtract fractions

0 = \frac{2 ( 3 ) - ( x + 4 ) ( x - 1 )}{3 ( x - 1 )}

0 = \frac{6 - ( x + 4 ) ( x - 1 )}{3 ( x - 1 )}

Multiply parentheses

0 = \frac{6 - ( x ^{2} - x + 4 x - 4 )}{3 ( x - 1 )}

Distribute $- 1$

0 = \frac{6 - x ^{2} + x - 4 x + 4}{3 ( x - 1 )}

Add and subtract terms

0 = \frac{- x ^{2} - 3 x + 1 0}{3 ( x - 1 )}

The fraction can be equal to

0

only if its numerator is equal to

0 .

This means that we need to solve the following equation.

- x^{2} - 3 x + 10 = 0

Let's first change the signs by multiplying both sides of the equation by

- 1 .

Then we will solve it by factoring.

- x^{2} - 3 x + 10 = 0

▼

Factor

$LHS \cdot (- 1) = RHS \cdot (- 1)$

x^{2} + 3 x - 10 = 0

Write as a difference

x^{2} + 5 x - 2 x - 10 = 0

Factor out $x & - 2$

x (x + 5) - 2 (x + 5) = 0

Factor out $x + 5$

(x - 2) (x + 5) = 0

According to the Zero Product Property, at least one of the parentheses should be equal to

0 .

x - 2 = 0 x + 5 = 0 \Rightarrow x = 2 x = - 5

We got that the zeroes, or

x -

intercepts of the function are

- 5

and

2 .

The answer is D.

Exercise