Question 1

$\frac{x^{2}-10 x+21}{x^{2}+14 x+49} \div \frac{x^{2}-6 x+9}{x^{2}-2 x-35}$

Accepted Answer

A)  $-\frac{x+5}{x-3}$ 
B)  $\frac{(x-7)^{2}(x+5)}{(x+7)^{2}(x-3)}$ 
C)  $\frac{(x-7)^{2}(x+5)}{(x+7)^{2}}$ 
D)  $\frac{(x-3)^{3}}{(x+7)^{2}(x+5)}$ 
A)  $-\frac{x+5}{x-3}$ 
B)  $\frac{(x-7)^{2}(x+5)}{(x+7)^{2}(x-3)}$ 
C)  $\frac{(x-7)^{2}(x+5)}{(x+7)^{2}}$ 
D)  $\frac{(x-3)^{3}}{(x+7)^{2}(x+5)}$

Question 2

Find the domain of the rational function
-$r(x)=\frac{x^{2}-4}{x^{2}+6 x-27}$

Accepted Answer

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. 
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 3

Use the given graph of a rational function $\mathbf{r}(\mathrm{x})$ to solve the problem.
-Find r(3).

Accepted Answer

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

Question 4

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac{49-x^{2}}{x^{2}-15 x+56}$

Accepted Answer

A)  $-\frac{x+7}{x+8}$ 
B)  $-\frac{x-7}{x-8}$ 
C)  $\frac{x+7}{x-8}$ 
D)  $-\frac{x+7}{x-8}$ 
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 5

Identify the given function as a linear function, a 
-$f(x)=\frac{x^{2}-16}{x^{2}+8 x+16}$

Accepted Answer

A)  Quadratic 
B)  Rational 
C)  Linear 
A)  Quadratic 
B)  Rational 
C)  Linear

Question 6

Simplify the complex fraction.
-$\frac{\frac{1}{8}+\frac{1}{10}}{\frac{1}{8}-\frac{1}{10}}$

Accepted Answer

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

Question 7

$\frac{4 x^{2}-7 x-5}{x^{2}+5 x}-\frac{3 x^{2}-2 x-5}{x^{2}+5 x}$

Accepted Answer

A)  $\frac{7 x^{2}-9 x-10}{x(x+5)}$ 
B)  1 
C)  $\frac{x-5}{x+5}$ 
D)  $\frac{x+5}{x-5}$ 
A)  $\frac{7 x^{2}-9 x-10}{x(x+5)}$ 
B)  1 
C)  $\frac{x-5}{x+5}$ 
D)  $\frac{x+5}{x-5}$

Question 8

$\frac{x+5}{x^{2}+9 x+14}+\frac{5 x-4}{x^{2}+6 x-7}$

Accepted Answer

A)  $6 x+1$ 
B)  $\frac{6 x^{2}+10 x-13}{(x+7)(x+2)(x-1)}$ 
C)  $\frac{6 x^{2}+10 x-13}{(x-7)(x-2)(x+1)}$ 
D)  $\frac{6 x+1}{2 x^{2}+15 x+7}$ 
A)  $6 x+1$ 
B)  $\frac{6 x^{2}+10 x-13}{(x+7)(x+2)(x-1)}$ 
C)  $\frac{6 x^{2}+10 x-13}{(x-7)(x-2)(x+1)}$ 
D)  $\frac{6 x+1}{2 x^{2}+15 x+7}$

Question 9

$\frac{4 x^{2}-11 x+3}{x^{2}+12 x+32}-\frac{3 x^{2}-11 x+19}{x^{2}+12 x+32}$

Accepted Answer

A)  $\frac{x+4}{x+8}$ 
B)  $\frac{7 x^{2}-22 x+22}{(x+8)(x+4)}$ 
C)  $\frac{x-4}{x+8}$ 
D)  $\frac{x^{2}-16}{(x+8)(x-4)}$ 
A)  $\frac{x+4}{x+8}$ 
B)  $\frac{7 x^{2}-22 x+22}{(x+8)(x+4)}$ 
C)  $\frac{x-4}{x+8}$ 
D)  $\frac{x^{2}-16}{(x+8)(x-4)}$

Question 10

$\frac{3 \mathrm{x}+3}{\mathrm{x}}+\frac{5 \mathrm{x}-1}{3 \mathrm{x}}$

Accepted Answer

A)  $\frac{14 x+8}{3 x^{2}}$ 
B)  $\frac{14 x+8}{3 x}$ 
C)  $\frac{4 x+8}{3 x}$ 
D)  $\frac{14 x-10}{3 x}$ 
A)  $\frac{14 x+8}{3 x^{2}}$ 
B)  $\frac{14 x+8}{3 x}$ 
C)  $\frac{4 x+8}{3 x}$ 
D)  $\frac{14 x-10}{3 x}$

Question 11

The sum of a number and 8 times its reciprocal is -6 . Find the number.

Accepted Answer

A)  2 or 4 
B)  -2 or -4 
C)  -2 
D)  -5 or 1 
A)  2 or 4 
B)  -2 or -4 
C)  -2 
D)  -5 or 1

Question 12

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac{15 x^{2}-105 x+180}{25 x^{2}-225 x+500}$

Accepted Answer

A)  $\frac{3(x+3)}{5(x+5)}$ 
B)  $\frac{3(x-3)}{5(x-5)}$ 
C)  $\frac{12}{25}$ 
D)  $\frac{9}{25}$ 
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 13

According to Ohm's law, the electric current, in amperes, in a circuit varies directly as the voltage. When 14 volts are applied, the current is 4 amperes. What is the current when 14 volts are applied?

Accepted Answer

A)  3.5 amperes 
B)  4 amperes 
C)  32 amperes 
D)  49 amperes 
A)  3.5 amperes 
B)  4 amperes 
C)  32 amperes 
D)  49 amperes

Question 14

$\frac{3 \mathrm{x}+12}{\mathrm{x}^{2}-3 \mathrm{x}-18}-\frac{\mathrm{x}+6}{\mathrm{x}^{2}-3 \mathrm{x}-18}$

Accepted Answer

A)  $2 x+6$ 
B)  $\frac{2}{x-6}$ 
C)  $\frac{2 x+18}{x^{2}-3 x-18}$ 
D)  $\frac{1}{x-3}$ 
A)  $2 x+6$ 
B)  $\frac{2}{x-6}$ 
C)  $\frac{2 x+18}{x^{2}-3 x-18}$ 
D)  $\frac{1}{x-3}$

Question 15

$\frac{4}{\mathrm{x}+2}-\frac{6}{\mathrm{x}-2}=\frac{4}{\mathrm{x}^{2}-4}$

Accepted Answer

A)  $\{12\}$ 
B)  $\{-12\}$. 
C)  $\{24\}$ 
D)  $\varnothing$ 
A)  $\{12\}$ 
B)  $\{-12\}$. 
C)  $\{24\}$ 
D)  $\varnothing$

Question 16

For the given rational functions $f(x)$ and $g(x), f(x)$ - $g(x)$.
-$f(x)=\frac{6}{x-4}, g(x)=\frac{25}{x-4}$

Accepted Answer

A)  $-\frac{19}{x-4}$ 
B)  $\frac{31}{x-4}$ 
C)  $\frac{19}{x-4}$ 
D)  $-\frac{19}{\mathrm{x}}$ 
A)  $-\frac{19}{x-4}$ 
B)  $\frac{31}{x-4}$ 
C)  $\frac{19}{x-4}$ 
D)  $-\frac{19}{\mathrm{x}}$

Question 17

For the given rational functions $f(x)$ and $g(x), f(x)+g(x)$.
-$f(x)=\frac{x}{x^{2}-3 x-10}, g(x)=\frac{2}{x^{2}-3 x-10}$

Accepted Answer

A)  $\frac{x-2}{x^{2}-3 x-10}$ 
B)  $\frac{1}{x-5}$ 
C)  $\frac{1}{x+2}$ 
D)  $\frac{1}{x^{2}-3 x-10}$ 
A)  $\frac{x-2}{x^{2}-3 x-10}$ 
B)  $\frac{1}{x-5}$ 
C)  $\frac{1}{x+2}$ 
D)  $\frac{1}{x^{2}-3 x-10}$

Question 18

Find the missing numerator and denominator.
-$\frac{x^{2}-81}{x} \div \frac{?}{?}=\frac{(x-9)(x-4)}{x}$

Accepted Answer

A)  $\frac{x+9}{x-4}$ 
B)  $\frac{x}{x-4}$ 
C)  $\frac{x}{(x-9)(x-4)}$ 
D)  $\frac{(x+9)\left(x^{2}+9\right)}{x(x-4)}$ 
A)  $\frac{x+9}{x-4}$ 
B)  $\frac{x}{x-4}$ 
C)  $\frac{x}{(x-9)(x-4)}$ 
D)  $\frac{(x+9)\left(x^{2}+9\right)}{x(x-4)}$

Question 19

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac{x^{2}-5 x-14}{x^{2}-7 x-18}$

Accepted Answer

A)  $\frac{-5 x-14}{-7 x-18}$ 
B)  $\frac{x-7}{x-9}$ 
C)  $\frac{-5 x-7}{-7 x-9}$ 
D)  $-\frac{x^{2}-5 x-14}{x^{2}-7 x-18}$ 
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 20

Find the domain of the rational function
-$r(x)=\frac{x^{2}-3 x}{x^{2}-4 x-12}$

Accepted Answer

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. 
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.

$\frac{x^{2}-10 x+21}{x^{2}+14 x+49} \div \frac{x^{2}-6 x+9}{x^{2}-2 x-35}$

Find the domain of the rational function - $r(x)=\frac{x^{2}-4}{x^{2}+6 x-27}$

Use the given graph of a rational function $\mathbf{r}(\mathrm{x})$ to solve the problem. -Find r(3). $Use the given graph of a rational function \mathbf{r}(\mathrm{x}) to solve the problem. -Find r(3).$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{49-x^{2}}{x^{2}-15 x+56}$

Identify the given function as a linear function, a - $f(x)=\frac{x^{2}-16}{x^{2}+8 x+16}$

Simplify the complex fraction. - $\frac{\frac{1}{8}+\frac{1}{10}}{\frac{1}{8}-\frac{1}{10}}$

$\frac{4 x^{2}-7 x-5}{x^{2}+5 x}-\frac{3 x^{2}-2 x-5}{x^{2}+5 x}$

$\frac{x+5}{x^{2}+9 x+14}+\frac{5 x-4}{x^{2}+6 x-7}$

$\frac{4 x^{2}-11 x+3}{x^{2}+12 x+32}-\frac{3 x^{2}-11 x+19}{x^{2}+12 x+32}$

$\frac{3 \mathrm{x}+3}{\mathrm{x}}+\frac{5 \mathrm{x}-1}{3 \mathrm{x}}$

The sum of a number and 8 times its reciprocal is -6 . Find the number.

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{15 x^{2}-105 x+180}{25 x^{2}-225 x+500}$

According to Ohm's law, the electric current, in amperes, in a circuit varies directly as the voltage. When 14 volts are applied, the current is 4 amperes. What is the current when 14 volts are applied?

$\frac{3 \mathrm{x}+12}{\mathrm{x}^{2}-3 \mathrm{x}-18}-\frac{\mathrm{x}+6}{\mathrm{x}^{2}-3 \mathrm{x}-18}$

$\frac{4}{\mathrm{x}+2}-\frac{6}{\mathrm{x}-2}=\frac{4}{\mathrm{x}^{2}-4}$

For the given rational functions $f(x)$ and $g(x), f(x)$ - $g(x)$ . - $f(x)=\frac{6}{x-4}, g(x)=\frac{25}{x-4}$

For the given rational functions $f(x)$ and $g(x), f(x)+g(x)$ . - $f(x)=\frac{x}{x^{2}-3 x-10}, g(x)=\frac{2}{x^{2}-3 x-10}$

Find the missing numerator and denominator. - $\frac{x^{2}-81}{x} \div \frac{?}{?}=\frac{(x-9)(x-4)}{x}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{x^{2}-5 x-14}{x^{2}-7 x-18}$

Find the domain of the rational function - $r(x)=\frac{x^{2}-3 x}{x^{2}-4 x-12}$

Review of Real Numbers

Linear and Absolute Value Equations and Inequalities

Graphing Linear Equations

Systems of Equations

Exponents, Polynomials, and Factoring Polynomials

Radical Expressions and Equations

Quadratic Equations

Logarithmic and Exponential Functions

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Exam 6: Rational Expressions and Equations

x2−10x+21x2+14x+49÷x2−6x+9x2−2x−35\frac{x^{2}-10 x+21}{x^{2}+14 x+49} \div \frac{x^{2}-6 x+9}{x^{2}-2 x-35}x2+14x+49x2−10x+21​÷x2−2x−35x2−6x+9​

Find the domain of the rational function - r(x)=x2−4x2+6x−27r(x)=\frac{x^{2}-4}{x^{2}+6 x-27}r(x)=x2+6x−27x2−4​

Use the given graph of a rational function r(x)\mathbf{r}(\mathrm{x})r(x) to solve the problem. -Find r(3).

Simplify the rational expression. (Assume the denominator is nonzero.) - 49−x2x2−15x+56\frac{49-x^{2}}{x^{2}-15 x+56}x2−15x+5649−x2​

Identify the given function as a linear function, a - f(x)=x2−16x2+8x+16f(x)=\frac{x^{2}-16}{x^{2}+8 x+16}f(x)=x2+8x+16x2−16​

Simplify the complex fraction. - 18+11018−110\frac{\frac{1}{8}+\frac{1}{10}}{\frac{1}{8}-\frac{1}{10}}81​−101​81​+101​​

4x2−7x−5x2+5x−3x2−2x−5x2+5x\frac{4 x^{2}-7 x-5}{x^{2}+5 x}-\frac{3 x^{2}-2 x-5}{x^{2}+5 x}x2+5x4x2−7x−5​−x2+5x3x2−2x−5​

x+5x2+9x+14+5x−4x2+6x−7\frac{x+5}{x^{2}+9 x+14}+\frac{5 x-4}{x^{2}+6 x-7}x2+9x+14x+5​+x2+6x−75x−4​

4x2−11x+3x2+12x+32−3x2−11x+19x2+12x+32\frac{4 x^{2}-11 x+3}{x^{2}+12 x+32}-\frac{3 x^{2}-11 x+19}{x^{2}+12 x+32}x2+12x+324x2−11x+3​−x2+12x+323x2−11x+19​

3x+3x+5x−13x\frac{3 \mathrm{x}+3}{\mathrm{x}}+\frac{5 \mathrm{x}-1}{3 \mathrm{x}}x3x+3​+3x5x−1​

The sum of a number and 8 times its reciprocal is -6 . Find the number.

Simplify the rational expression. (Assume the denominator is nonzero.) - 15x2−105x+18025x2−225x+500\frac{15 x^{2}-105 x+180}{25 x^{2}-225 x+500}25x2−225x+50015x2−105x+180​

According to Ohm's law, the electric current, in amperes, in a circuit varies directly as the voltage. When 14 volts are applied, the current is 4 amperes. What is the current when 14 volts are applied?

3x+12x2−3x−18−x+6x2−3x−18\frac{3 \mathrm{x}+12}{\mathrm{x}^{2}-3 \mathrm{x}-18}-\frac{\mathrm{x}+6}{\mathrm{x}^{2}-3 \mathrm{x}-18}x2−3x−183x+12​−x2−3x−18x+6​

4x+2−6x−2=4x2−4\frac{4}{\mathrm{x}+2}-\frac{6}{\mathrm{x}-2}=\frac{4}{\mathrm{x}^{2}-4}x+24​−x−26​=x2−44​

For the given rational functions f(x)f(x)f(x) and g(x),f(x)g(x), f(x)g(x),f(x) - g(x)g(x)g(x) . - f(x)=6x−4,g(x)=25x−4f(x)=\frac{6}{x-4}, g(x)=\frac{25}{x-4}f(x)=x−46​,g(x)=x−425​

For the given rational functions f(x)f(x)f(x) and g(x),f(x)+g(x)g(x), f(x)+g(x)g(x),f(x)+g(x) . - f(x)=xx2−3x−10,g(x)=2x2−3x−10f(x)=\frac{x}{x^{2}-3 x-10}, g(x)=\frac{2}{x^{2}-3 x-10}f(x)=x2−3x−10x​,g(x)=x2−3x−102​

Find the missing numerator and denominator. - x2−81x÷??=(x−9)(x−4)x\frac{x^{2}-81}{x} \div \frac{?}{?}=\frac{(x-9)(x-4)}{x}xx2−81​÷??​=x(x−9)(x−4)​

Simplify the rational expression. (Assume the denominator is nonzero.) - x2−5x−14x2−7x−18\frac{x^{2}-5 x-14}{x^{2}-7 x-18}x2−7x−18x2−5x−14​

Find the domain of the rational function - r(x)=x2−3xx2−4x−12r(x)=\frac{x^{2}-3 x}{x^{2}-4 x-12}r(x)=x2−4x−12x2−3x​

Review of Real Numbers

Linear and Absolute Value Equations and Inequalities

Graphing Linear Equations

Systems of Equations

Exponents, Polynomials, and Factoring Polynomials

Radical Expressions and Equations

Quadratic Equations

Logarithmic and Exponential Functions

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

$\frac{x^{2}-10 x+21}{x^{2}+14 x+49} \div \frac{x^{2}-6 x+9}{x^{2}-2 x-35}$

Find the domain of the rational function - $r(x)=\frac{x^{2}-4}{x^{2}+6 x-27}$

Use the given graph of a rational function $\mathbf{r}(\mathrm{x})$ to solve the problem. -Find r(3). $Use the given graph of a rational function \mathbf{r}(\mathrm{x}) to solve the problem. -Find r(3).$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{49-x^{2}}{x^{2}-15 x+56}$

Identify the given function as a linear function, a - $f(x)=\frac{x^{2}-16}{x^{2}+8 x+16}$

Simplify the complex fraction. - $\frac{\frac{1}{8}+\frac{1}{10}}{\frac{1}{8}-\frac{1}{10}}$

$\frac{4 x^{2}-7 x-5}{x^{2}+5 x}-\frac{3 x^{2}-2 x-5}{x^{2}+5 x}$

$\frac{x+5}{x^{2}+9 x+14}+\frac{5 x-4}{x^{2}+6 x-7}$

$\frac{4 x^{2}-11 x+3}{x^{2}+12 x+32}-\frac{3 x^{2}-11 x+19}{x^{2}+12 x+32}$

$\frac{3 \mathrm{x}+3}{\mathrm{x}}+\frac{5 \mathrm{x}-1}{3 \mathrm{x}}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{15 x^{2}-105 x+180}{25 x^{2}-225 x+500}$

$\frac{3 \mathrm{x}+12}{\mathrm{x}^{2}-3 \mathrm{x}-18}-\frac{\mathrm{x}+6}{\mathrm{x}^{2}-3 \mathrm{x}-18}$

$\frac{4}{\mathrm{x}+2}-\frac{6}{\mathrm{x}-2}=\frac{4}{\mathrm{x}^{2}-4}$

For the given rational functions $f(x)$ and $g(x), f(x)$ - $g(x)$ . - $f(x)=\frac{6}{x-4}, g(x)=\frac{25}{x-4}$

For the given rational functions $f(x)$ and $g(x), f(x)+g(x)$ . - $f(x)=\frac{x}{x^{2}-3 x-10}, g(x)=\frac{2}{x^{2}-3 x-10}$

Find the missing numerator and denominator. - $\frac{x^{2}-81}{x} \div \frac{?}{?}=\frac{(x-9)(x-4)}{x}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{x^{2}-5 x-14}{x^{2}-7 x-18}$

Find the domain of the rational function - $r(x)=\frac{x^{2}-3 x}{x^{2}-4 x-12}$