Determine the values of x for which the given rational expression is undefined. — ACT

Determine the values of $x$ for which the given rational expression is undefined.

Answer:The rational expression

\frac{x + 5}{x^2 - 4x - 12}

is undefined for two real values of

x

. The smaller of these values is 【-2】 and the larger of these values is 【6】.

The rational expression is undefined for

x = -2

and

x = 6

The expression is undefined when the denominator is zero. Solving

x^2 - 4x - 12 = 0

yields

(x-6)(x+2) = 0

, which gives

x = -2

and

x = 6

. The smaller value is

-2

and the larger value is

6

Identify the condition that makes a rational expression undefined.

The denominator must equal zero:

x^2 - 4x - 12 = 0

Division by zero is undefined in the set of real numbers.

Factor the quadratic equation

x^2 - 4x - 12 = 0

(x - 6)(x + 2) = 0

We need to find two numbers that multiply to

-12

and add to

-4

. These numbers are

-6

and

2

Solve for

x

by setting each factor to zero.

x = 6

x = -2

By the zero product property, if

(x - 6)(x + 2) = 0

, then

x - 6 = 0

x + 2 = 0

Determine the smaller and larger values to place in the blanks.

Smaller value is

-2

and larger value is

6

Comparing the two numbers,

-2

is less than

6

A rational expression is undefined when its denominator equals zero.

Question