Deck 6: Rational Expressions and Equations

Question 1

Evaluate the rational expression for the given value of the variable.

- $\frac{x}{x-9}$ for $x=-9$

A) Undefined
B) 0
C) $-\frac{1}{9}$
D) $\frac{1}{2}$

Answer 1

$\frac{1}{2}$

Question 2

Evaluate the rational expression for the given value of the variable.

- $\frac{x-1}{4 x+6}$ for $x=-2$

A) $\frac{3}{8}$

B) $\frac{3}{20}$

C) $-\frac{3}{5}$

D) $\frac{3}{2}$

Answer 2

$\frac{3}{2}$

Question 3

Evaluate the rational expression for the given value of the variable.

- $\frac{2 x-2}{2 x^{2}+5 x-3}$ for $x=-4$

A) $-\frac{2}{3}$

B) $-\frac{10}{9}$

C) $-\frac{2}{11}$

D) $\frac{2}{3}$

Answer 3

$-\frac{10}{9}$

Question 4

Evaluate the rational expression for the given value of the variable.

- $\frac{x^{2}-5 x-15}{x^{2}-10 x+14}$ for $x=2$

A) -5

B) $\frac{23}{4}$

C) -2

D) $\frac{21}{2}$

Question 5

Evaluate the rational expression for the given value of the variable.

- $\frac{x^{2}-3 x}{x-4}$ for $x=9$

A) $\frac{54}{5}$

B) $-\frac{54}{5}$

C) 0

D) Undefined

Question 6

Find all values of the variable for which the rational expression is undefined.

- $\frac{13}{x-6}$

A) 13
B) 0
C) 6
D) -6

Question 7

Find all values of the variable for which the rational expression is undefined.

- $\frac{2}{x-4}$

A) 0
B) 4
C) -4
D) None

Question 8

Find all values of the variable for which the rational expression is undefined.

- $\frac{7}{x+8}$

A) 8
B) _ 8
C) 0
D) Nono

Question 9

Find all values of the variable for which the rational expression is undefined.

- $\frac{x-4}{7-x}$

A) 7,4
B) None
C) 7
D) -7

Question 10

Find all values of the variable for which the rational expression is undefined.

- $\frac{x-6}{6 x-24}$

A) 0
B) 4
C) 24
D) -6

Question 11

Find all values of the variable for which the rational expression is undefined.

- $\frac{x^{2}-36}{x^{2}-17 x+72}$

A) 8,9
B) $-9,8$
C) $-8,9$
D) $-8,-9$

Question 12

Find all values of the variable for which the rational expression is undefined.

- $\frac{x^{2}-16}{x^{2}-3 x-40}$

A) $-8,5$
B) 0
C) $-4,4$
D) $-5,8$

Question 13

Find all values of the variable for which the rational expression is undefined.

- $\frac{x^{2}+2 x-35}{x^{2}-64}$

A) $-5,7$
B) $-8,8$
C) -5
D) $-7,5$

Question 14

Find all values of the variable for which the rational expression is undefined.

- $\frac{81 \mathrm{x}}{\mathrm{x}^{2}-3 \mathrm{x}-18}$

A) $-9,9$
B) $-6,3$
C) $-3,6$
D) 0

Question 15

Evaluate the rational function.

- $r(x)=\frac{17}{x^{2}+2 x-15}, r(-9)$

A) $-\frac{17}{48}$
B) Undefined
C) $\frac{17}{48}$
D) $\frac{1}{24}$

Question 16

Evaluate the rational function.

- $r(x)=\frac{x}{x^{2}-9 x+20}, r(7)$

A) $-\frac{7}{6}$
B) $\frac{7}{6}$
C) Undefined
D) 0

Question 17

Evaluate the rational function.

- $r(x)=\frac{x^{2}+5 x-14}{x^{2}+4 x-12}, r(2)$

A) $-\frac{9}{8}$
B) $\frac{5}{4}$
C) $\frac{9}{8}$
D) Undefined

Question 18

Evaluate the rational function.

- $r(x)=\frac{x^{3}+3 x^{2}-14 x+10}{x^{2}+7 x-17}, r(2)$

A) -2
B) Undefined
C) 2
D) - 1

Question 19

Find the domain of the rational function

- $r(x)=\frac{x^{2}-10 x+24}{x^{2}-6 x}$

A) All real numbers except 0 and- 6 .
B) All real numbers except 0 and 6.
C) All real numbers except 0 .
D) All real numbers except - 6 and 6 .

Question 20

Find the domain of the rational function

- $r(x)=\frac{x^{2}+x-36}{x^{2}-17 x+72}$

A) All real numbers except - 9 and - 8 .
B) All real numbers except - 9 and 8.
C) All real numbers except - 8 and 9 .
D) All real numbers except 9 and 8.

Question 21

Find the domain of the rational function

- $r(x)=\frac{x^{2}-3 x}{x^{2}-4 x-12}$

A) All real numbers except -2 and 6 .
B) All real numbers except 0 and 3.
C) All real numbers except -6 and 2 .
D) All real numbers except - 3 and 3.

Question 22

Find the domain of the rational function

- $r(x)=\frac{x^{2}-4}{x^{2}+6 x-27}$

A) All real numbers except - 3 and 9 .
B) All real numbers except - 2 and 2.
C) All real numbers except - 9 and 3 .
D) All real numbers.

Question 23

Find the domain of the rational function

- $r(x)=\frac{x^{2}+4 x-32}{x^{2}-9}$

A) All real numbers except - 8 and 4 .
B) All real numbers except - 3 and 3.
C) All real numbers except -4 and 8.
D) All real numbers.

Question 24

Identify the given function as a linear function, a

- $f(x)=x^{2}+2 x-8$

A) Quadratic
B) Rational
C) Linear

Question 25

Identify the given function as a linear function, a

- $f(x)=4 x+2$

A) Rational
B) Quadratic
C) Linear

Question 26

Identify the given function as a linear function, a

- $f(x)=\frac{x^{2}}{x^{3}+3 x-28}$

A) Quadratic
B) Linear
C) Rational

Question 27

Identify the given function as a linear function, a

- $f(x)=\frac{x^{2}-16}{x^{2}+8 x+16}$

A) Quadratic
B) Rational
C) Linear

Question 28

Identify the given function as a linear function, a

- $f(x)=\frac{4}{9} x+\frac{1}{3}$

A) Quadratic
B) Rational
C) Linear

Question 29

Identify the given function as a linear function, a

- $f(x)=\frac{1}{3} x^{2}-25$

A) Linear
B) Quadratic
C) Rational

Question 30

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{18 x^{3}}{6 x}$

A) 12
B) $12 x^{2}$
C) $3 x$
D) $3 x^{2}$

Question 31

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{(x+2)(x-2)}{x+2}$

A) $x-2$
B) - $(x+2)$
C) $\frac{x^{2}-4}{x+2}$
D) $x+2$

Question 32

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{(\mathrm{x}+2)(\mathrm{x}-4)}{(\mathrm{x}-4)(\mathrm{x}+7)}$

A) $\frac{x-7}{x-2}$

B) $\frac{x-2}{x-7}$

C) $\frac{x+2}{x+7}$

D) $\frac{2 x-4}{2 x+3}$

Question 33

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{x^{2}+2 x-15}{x^{2}+6 x-27}$

A) $\frac{2 x-5}{6 x-9}$

B) $\frac{x+5}{x+9}$

C) $-\frac{x^{2}+2 x-15}{x^{2}+6 x-27}$

D) $\frac{2 x-15}{6 x-27}$

Question 34

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{x^{2}+6 x-27}{x^{2}+18 x+81}$

A) $\frac{6 x+27}{18 x-18}$

B) $\frac{x-3}{x+9}$

C) $-\frac{x^{2}+6 x-27}{x^{2}+18 x+81}$

D) $\frac{6 x-27}{18 x+81}$

Question 35

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{x^{2}-5 x-14}{x^{2}-7 x-18}$

A) $\frac{-5 x-14}{-7 x-18}$

B) $\frac{x-7}{x-9}$

C) $\frac{-5 x-7}{-7 x-9}$

D) $-\frac{x^{2}-5 x-14}{x^{2}-7 x-18}$

Question 36

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{x^{2}-64}{x+8}$

A) $x-8$
B) $x-64$
C) $x+8$
D) $\frac{1}{x-8}$

Question 37

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{x^{2}-2 x}{(x+3)(x-2)}$

A) $\frac{x-2}{x+3}$

B) $\frac{x^{2}}{x+3}$

C) $\frac{1}{x+3}$

D) $\frac{x}{x+3}$

Question 38

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{x^{2}-49}{(x+7)^{2}}$

A) $\frac{x+7}{x-7}$

B) $x-7$

C) $\frac{x^{2}-49}{(x+7)^{2}}$

D) $\frac{x-7}{x+7}$

Question 39

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{15 x^{2}-105 x+180}{25 x^{2}-225 x+500}$

A) $\frac{3(x+3)}{5(x+5)}$

B) $\frac{3(x-3)}{5(x-5)}$

C) $\frac{12}{25}$

D) $\frac{9}{25}$

Question 40

Determine whether the two binomials are an opposites.

- $x-5$ and $5-x$

Question 41

Determine whether the two binomials are an opposites.

- $x+8$ and $8+x$

Question 42

Determine whether the two binomials are an opposites.

- $2 x+3$ and $-2 x+3$

Question 43

Determine whether the two binomials are an opposites.

- $2 x-3$ and $3 x-2$

Question 44

Determine whether the two binomials are an opposites.

- $3 x-6$ and $3 x+6$

Question 45

Determine whether the two binomials are an opposites.

-8 - $x$ and $x-8$

Question 46

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{7 x-35}{(3+x)(5-x)}$

A) $-\frac{7}{3+x}$

B) $-\frac{7 x-35}{3+x}$

C) $\frac{7}{3+x}$

D) $\frac{7 x-35}{3+x}$

Question 47

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{2 x^{2}-50}{20-4 x}$

A) $\frac{x+5}{2}$

B) $\frac{x^{2}-25}{5-x}$

C) $-\frac{x+5}{2}$

D) $-2(x+5)$

Question 48

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{(\mathrm{x}-8)(\mathrm{x}+5)}{64-\mathrm{x}^{2}}$

A) $\frac{x+5}{8+x}$

B) $-\frac{x+5}{8+x}$

C) $\frac{x-5}{8-x}$

D) $-\frac{x-5}{8-x}$

Question 49

Simplify the rational expression. (Assume the denominator is nonzero.)

- $\frac{49-x^{2}}{x^{2}-15 x+56}$

A) $-\frac{x+7}{x+8}$

B) $-\frac{x-7}{x-8}$

C) $\frac{x+7}{x-8}$

D) $-\frac{x+7}{x-8}$

Question 50

Use the given graph of a rational function $\mathbf{r}(\mathrm{x})$ to solve the problem.

-Find r(3).

A) -1
B) -3
C) 0
D) 1

Question 51

Find $r(-2)$ .

A) 1
B) 0
C) -1
D) 3

Question 52

Find $r(0)$ .

A) 2
B) 1
C) 0
D) -2

Question 53

Find all values $x$ such that $r(x)=3$ .

A) -1
B) 0
C) 1
D) -3

Question 54

Find all values $x$ such that $r(x)=0$ .

A) -5
B) 5
C) -3
D) 3

Question 55

Find all values $x$ such that $r(x)=0$ .

A) 0
B) -3
C) $-3,3$
D) 3
Find any vertical asymptotes as well as the location of any holes in the graph of the given rational function.

Question 56

Find any vertical asymptotes as well as the location of any holes in the graph of the given rational function.

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

A) vertical asymptote: $\mathrm{x}=2$ ; holes: none
B) vertical asymptote: $\mathrm{x}=3$ ; holes: none
C) vertical asymptote: $\mathrm{x}=-2$ ; holes: none
D) vertical asymptote: $\mathrm{x}=-3$ ; holes: none

Question 57

Find any vertical asymptotes as well as the location of any holes in the graph of the given rational function.

- $f(x)=\frac{x^{2}-49}{x^{2}+7 x+10}$

A) vertical asymptotes: $x=-7, x=7$ ; holes: none
B) vertical asymptotes: $x=-5, x=-2$ ; holes: none
C) vertical asymptotes: $x=-5, x=-7$ ; hole: $(7,0)$
D) vertical asymptotes: $x=5, x=2$ ; hole: $(7,0)$

Question 58

Find any vertical asymptotes as well as the location of any holes in the graph of the given rational function.

- $f(x)=\frac{x^{2}-5 x-36}{x-9}$

A) vertical asymptotes: none; hole: $(9,0)$
B) vertical asymptotes: none; hole: $(9,13)$
C) vertical asymptotes: $x=-4, x=9$ ; hole: $(-4,-13)$
D) vertical asymptote: $x=9$ ; holes: none

Question 59

$\frac{x+4}{x-3} \cdot \frac{x-6}{x+3}$

A) $\frac{(x-3)(x+3)}{(x+4)(x-6)}$

B) $\frac{(\mathrm{x}-3)(\mathrm{x}-6)}{(\mathrm{x}+4)(\mathrm{x}+3)}$

C) $\frac{(x+4)(x-6)}{(x-3)(x+3)}$

D) $\frac{(\mathrm{x}+4)(\mathrm{x}+3)}{(\mathrm{x}-3)(\mathrm{x}-6)}$

Question 60

$\frac{(\mathrm{x}-2)(\mathrm{x}+3)}{(\mathrm{x}+7)(\mathrm{x}-2)} \cdot \frac{(\mathrm{x}-10)(\mathrm{x}+7)}{(\mathrm{x}+3)(\mathrm{x}+6)}$

A) $\frac{x-10}{x+6}$

B) $-\frac{x-10}{x+6}$

C) $\frac{x+3}{x-2}$

D) $\frac{x+10}{x-6}$

Question 61

$\frac{x^{2}-1}{3 x+3} \cdot \frac{x+3}{x-1}$

A) $\frac{x+3}{3}$

B) $x+3$

C) $\frac{1}{3}$

D) $\frac{(x+3)(x-1)}{3(x-1)}$

Question 62

$\frac{x-6}{(x+2)^{2}} \cdot \frac{x^{2}-4 x-12}{(x-6)^{2}}$

A) $\frac{1}{x+2}$

B) $\frac{x-6}{x+2}$

C) $\frac{1}{x-6}$

D) $x+2$

Question 63

$\frac{x^{2}+18 x+81}{x^{2}-49} \cdot \frac{x+7}{x+9}$

A) $\frac{x+9}{x+7}$

B) $\frac{x+9}{x-7}$

C) $\frac{x+9}{(x-7)^{2}}$

D) $\frac{x-9}{x-7}$

Question 64

$\frac{4 x+8}{x-3} \cdot \frac{2 x^{2}-12 x+18}{x^{2}-4}$

A) $\frac{8(x+2)(x-3)}{x^{2}-4}$

B) $\frac{4(x-3)}{x-2}$

C) $\frac{8(x-3)}{x-2}$

D) $\frac{8(x-3)}{x+2}$

Question 65

$\frac{6 x-24}{16-x^{2}} \cdot \frac{5 x^{2}+6 x+1}{30 x+6}$

A) $\frac{x+1}{x+4}$

B) $-\frac{6(x+1)}{x+4}$

C) $\frac{x+1}{4-x}$

D) $-\frac{x+1}{x+4}$

Question 66

$\frac{x^{2}+5 x+4}{x^{2}+10 x+24} \cdot \frac{x^{2}-11 x+30}{x^{2}-4 x-5}$

A) -1

B) $\frac{x-6}{x+6}$

C) $\frac{x+6}{x-6}$

D) 1

Question 67

$\frac{x^{2}+12 x+27}{x^{2}+18 x+81} \cdot \frac{x^{2}+9 x}{x^{2}-3 x-18}$

A) $\frac{x}{x-6}$

B) $\frac{1}{x-6}$

C) $\frac{x}{x^{2}+18 x+81}$

D) $\frac{x^{2}+9 x}{x-6}$

Question 68

$\frac{x^{2}+8 x+15}{x^{2}+10 x+21} \cdot \frac{x^{2}+9 x+14}{x^{2}+7 x+10}$

A) $\frac{x+5}{x+7}$

B) 1

C) $\frac{x+7}{x+2}$

D) $\frac{1}{x+2}$

Question 69

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ .

- $f(x)=\frac{8}{(x-8)}, g(x)=\frac{x^{2}-4 x-32}{(x+4)^{2}}$

A) $\frac{8(x-8)}{x+4}$

B) $8(x+4)$

C) $\frac{8}{x+4}$

D) $\frac{8}{x-8}$

Question 70

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ .

- $f(x)=x+8, g(x)=\frac{x^{2}+5 x-24}{4 x^{2}+24 x-64}$

A) $\frac{x-3}{4 x-8}$

B) $\frac{(x+8)(x-3)}{4 x-8}$

C) $\frac{(x+8)^{2}(x-3)}{4 x-8}$

D) $\frac{(x-8)(x-3)}{4 x+8}$

Question 71

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ .

- $f(x)=\frac{x^{2}+5 x+6}{x^{2}+8 x+12}, g(x)=\frac{x^{2}-5 x-6}{x^{2}+4 x+3}$

A) $\frac{x-6}{x+6}$

B) -1

C) $\frac{x+6}{x-6}$

D) 1

Question 72

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ .

- $f(x)=\frac{x^{2}+9 x+18}{x^{2}+12 x+27}, g(x)=\frac{x^{2}+9 x}{x^{2}+3 x-18}$

A) $\frac{x}{x^{2}+12 x+27}$

B) $\frac{x}{x-3}$

C) $\frac{x^{2}+9 x}{x-3}$

D) $\frac{1}{x-3}$

Question 73

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ .

- $f(x)=\frac{x^{2}+5 x+6}{x^{2}+8 x+15}, g(x)=\frac{x^{2}+5 x}{x^{2}+10 x+16}$

A) $\frac{x}{x^{2}+8 x+15}$

B) $\frac{x^{2}+5 x}{x+8}$

C) $\frac{x}{x+8}$

D) $\frac{1}{x+8}$

Question 74

Find the missing numerator and denominator.

- $\frac{x+4}{x-7} \cdot \frac{?}{?}=\frac{(x+4)(x-1)}{(x-7)(x+3)}$

A) $\frac{x-1}{x+3}$

B) $\frac{x(x-1)}{x(x+3)}$

C) $\frac{x+1}{x-3}$

D) $\frac{x(4 x-1)}{x(7 x+3)}$

Question 75

Find the missing numerator and denominator.

- $\frac{x^{2}+18 x+81}{x^{2}-16} \cdot \frac{?}{?}=\frac{x+9}{x+4}$

A) $\frac{x+9}{x-4}$

B) $\frac{x-9}{x+4}$

C) $\frac{x-4}{x+9}$

D) $\frac{x+9}{(x+4)^{2}}$

Question 76

Find the missing numerator and denominator.

- $\frac{\mathrm{x}^{2}}{\mathrm{x}^{2}+5 \mathrm{x}-36} \cdot \frac{?}{?}=\frac{\mathrm{x}(\mathrm{x}+4)}{4(\mathrm{x}+9)}$

A) $\frac{x^{2}+16}{4 x}$

B) $\frac{x(x-4)}{4(x-9)}$

C) $\frac{x+4}{4 x(x-9)}$

D) $\frac{x^{2}-16}{4 x}$

Question 77

Find the missing numerator and denominator.

- $\frac{(\mathrm{x}-2)(\mathrm{x}-6)}{(\mathrm{x}-1)(\mathrm{x}-9)} \cdot \frac{?}{?}=\frac{(\mathrm{x}-6)(\mathrm{x}-3)}{(\mathrm{x}-1)(\mathrm{x}-4)}$

A) $\frac{(x+3)(x-9)}{(x+4)(x-2)}$

B) $\frac{(x+3)(x+9)}{(x+4)(x+2)}$

C) $\frac{(x-3)(x-9)}{(x-4)(x-2)}$

D) $\frac{(x-3)(x+9)}{(x-4)(x+2)}$

Question 78

$\frac{x+3}{x-4} \div \frac{x+7}{x-7}$

A) $\frac{(x+3)(x-7)}{(x-4)(x+7)}$

B) $\frac{(\mathrm{x}-4)(\mathrm{x}+7)}{(\mathrm{x}+3)(\mathrm{x}-7)}$

C) $\frac{(x+3)(x+7)}{(x-4)(x-7)}$

D) $\frac{(\mathrm{x}-4)(\mathrm{x}-7)}{(\mathrm{x}+3)(\mathrm{x}+7)}$

Question 79

$\frac{x^{2}-1}{x} \div \frac{x-1}{x+6}$

A) $\frac{x}{(x+1)(x+6)}$

B) $\frac{(x-1)\left(x^{2}-1\right)}{x(x+6)}$

C) $\frac{(x+1)(x+6)}{x}$

D) $(x+1)(x+6)$

Question 80

$\frac{(2 \mathrm{x}-5)(\mathrm{x}+2)}{(\mathrm{x}+8)(\mathrm{x}-3)} \div \frac{(\mathrm{x}+2)(3 \mathrm{x}+5)}{(\mathrm{x}+8)(\mathrm{x}-3)}$

A) $-\frac{2 x-5}{3 x+5}$

B) $\frac{x-8}{x+2}$

C) $\frac{2}{3}$

D) $\frac{2 x-5}{3 x+5}$