Question 1

Find the missing numerator and denominator.
-$\frac{x^{2}-1}{x^{2}+6 x+9} \div \frac{?}{?}=\frac{(x+1)(x-5)}{5(x+3)}$

Accepted Answer

A)  $\frac{5 x+5}{x^{2}-2 x-15}$ 
B)  $\frac{x+3}{(x+1)(x-5)}$ 
C)  $\frac{(x+1)(x-5)}{x+3}$ 
D)  $\frac{5 x-5}{x^{2}-2 x-15}$ 
A)  $\frac{5 x+5}{x^{2}-2 x-15}$ 
B)  $\frac{x+3}{(x+1)(x-5)}$ 
C)  $\frac{(x+1)(x-5)}{x+3}$ 
D)  $\frac{5 x-5}{x^{2}-2 x-15}$

Question 2

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

Accepted Answer

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

For the given rational functions $f(x)$ and $g(x), f(x)+g(x)$.
-$f(x)=\frac{x}{16-x^{2}}, g(x)=\frac{3}{8 x-32}$

Accepted Answer

A)  $\frac{x+3}{8(x+4)(x-4)}$ 
B)  $\frac{-5 x-12}{8(x+4)(x-4)}$ 
C)  $\frac{5 x-12}{8(x+4)(x-4)}$ 
D)  $\frac{-5 x+12}{8(x+4)(x-4)}$ 
A)  $\frac{x+3}{8(x+4)(x-4)}$ 
B)  $\frac{-5 x-12}{8(x+4)(x-4)}$ 
C)  $\frac{5 x-12}{8(x+4)(x-4)}$ 
D)  $\frac{-5 x+12}{8(x+4)(x-4)}$

Question 4

$\frac{3 x}{x-4}-\frac{12}{x-4}$

Accepted Answer

A)  3 
B)  $\frac{3}{x-4}$ 
C)  0 
D)  $\frac{3(x+4)}{x-4}$ 
A)  3 
B)  $\frac{3}{x-4}$ 
C)  0 
D)  $\frac{3(x+4)}{x-4}$

Question 5

For the given functions $f(x)$ and $g(x)$, find $f(x) \div g(x)$.
-$f(x)=\frac{x^{2}+8 x+15}{x^{2}+12 x+35}, g(x)=\frac{x^{2}+3 x}{x^{2}-2 x-63}$

Accepted Answer

A)  $\frac{x}{x^{2}+12 x+35}$ 
B)  $\frac{x-9}{x^{2}+7 x}$ 
C)  $x-9$ 
D)  $\frac{x-9}{x}$ 
A)  $\frac{x}{x^{2}+12 x+35}$ 
B)  $\frac{x-9}{x^{2}+7 x}$ 
C)  $x-9$ 
D)  $\frac{x-9}{x}$

Question 6

$\frac{8 x+4 y}{5 x^{2} y}-\frac{3 x-3 y}{x y^{2}}$

Accepted Answer

A)  $\frac{8 x^{2} y^{2}+4 x y^{3}-15 x^{3} y+15 x^{2} y^{2}}{5 x^{3} y^{3}}$ 
B)  $\frac{15 x^{2}-7 x y+4 y^{2}}{5 x^{2} y^{2}}$ 
C)  $\frac{5 x+7 y}{5 x^{2} y^{2}}$ 
D)  $\frac{-15 x^{2}+23 x y+4 y^{2}}{5 x^{2} y^{2}}$ 
A)  $\frac{8 x^{2} y^{2}+4 x y^{3}-15 x^{3} y+15 x^{2} y^{2}}{5 x^{3} y^{3}}$ 
B)  $\frac{15 x^{2}-7 x y+4 y^{2}}{5 x^{2} y^{2}}$ 
C)  $\frac{5 x+7 y}{5 x^{2} y^{2}}$ 
D)  $\frac{-15 x^{2}+23 x y+4 y^{2}}{5 x^{2} y^{2}}$

Question 7

For the given functions $f(x)$ and $g(x)$, find $f(x) \div g(x)$.
-$f(x)=(x+6), g(x)=\frac{3 x^{2}+12 x-36}{x^{2}+7 x+6}$

Accepted Answer

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

Question 8

A plane flies 400 miles with the wind and 350 miles against the wind in the same length of time. If the speed of the wind is $25 \mathrm{mph}$, what is the speed of the plane in calm air?

Accepted Answer

A)  $400 \mathrm{mph}$ 
B)  $365 \mathrm{mph}$ 
C)  $380 \mathrm{mph}$ 
D)  $375 \mathrm{mph}$ 
A)  $400 \mathrm{mph}$ 
B)  $365 \mathrm{mph}$ 
C)  $380 \mathrm{mph}$ 
D)  $375 \mathrm{mph}$

Question 9

Newton's Law of universal gravitation relates the force $F$ with which a particle of mass $m_{1}$ attracts a particle of mass $\mathrm{m}_{2}$ to the gravitational constant $\mathrm{G}$ and the distance $r$ between the two objects: $\mathrm{F}=\frac{\mathrm{Gm} 1 \mathrm{~m} 2}{\mathrm{r}^{2}}$. Solve this equation for $\mathrm{G}$.

Accepted Answer

A)  $\mathrm{G}=\frac{\mathrm{Fr}^{2}}{\mathrm{~m}_{1} \mathrm{~m}_{2}}$ 
B)  $G=\frac{r^{2} m_{1}}{F m 2}$ 
C)  $\mathrm{G}=\frac{\mathrm{Fm} 1}{\mathrm{r}^{2} \mathrm{~m}_{2}}$ 
D)  $\mathrm{G}=\frac{\mathrm{m}_{1} \mathrm{~m}_{2}}{\mathrm{Fr}^{2}}$ 
A)  $\mathrm{G}=\frac{\mathrm{Fr}^{2}}{\mathrm{~m}_{1} \mathrm{~m}_{2}}$ 
B)  $G=\frac{r^{2} m_{1}}{F m 2}$ 
C)  $\mathrm{G}=\frac{\mathrm{Fm} 1}{\mathrm{r}^{2} \mathrm{~m}_{2}}$ 
D)  $\mathrm{G}=\frac{\mathrm{m}_{1} \mathrm{~m}_{2}}{\mathrm{Fr}^{2}}$

Question 10

$\frac{\mathrm{x}}{\mathrm{x}^{2}+6 \mathrm{x}-27}+\frac{9}{\mathrm{x}^{2}+6 \mathrm{x}-27}$

Accepted Answer

A)  $\frac{1}{x+3}$ 
B)  $\frac{1}{x-3}$ 
C)  $\frac{x+9}{2\left(x^{2}+6 x-27\right)}$ 
D)  $\frac{x+9}{x^{2}+6 x-27}$ 
A)  $\frac{1}{x+3}$ 
B)  $\frac{1}{x-3}$ 
C)  $\frac{x+9}{2\left(x^{2}+6 x-27\right)}$ 
D)  $\frac{x+9}{x^{2}+6 x-27}$

Question 11

One number is one larger than another positive number. If the reciprocal of the smaller number is multiplied by the reciprocal of the larger number, the product is $\frac{1}{30}$. Find the two numbers.

Accepted Answer

A)  4,5 
B)  6,7 
C)  5, 6 
D)  3,4 
A)  4,5 
B)  6,7 
C)  5, 6 
D)  3,4

Question 12

A number minus 12 times its reciprocal is 1 . Find the number.

Accepted Answer

A)  4 or -3 
B)  -4 or 3 
C)  4 or 3 
D)  4 
A)  4 or -3 
B)  -4 or 3 
C)  4 or 3 
D)  4

Question 13

$\frac{x}{4}-\frac{x}{7}=5$

Accepted Answer

A)  $\{35\}$ 
B)  $\{20\}$ 
C)  $\left\{\frac{140}{3}\right\}$ 
D)  $\{28\}$ 
A)  $\{35\}$ 
B)  $\{20\}$ 
C)  $\left\{\frac{140}{3}\right\}$ 
D)  $\{28\}$

Question 14

$\frac{x^{2}+14 x+13}{x^{2}+5 x+6}+\frac{x^{2}-8 x-9}{x^{2}+5 x+6}$

Accepted Answer

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

Question 15

Find the missing numerator and denominator.
-$\frac{\mathrm{x}^{2}}{\mathrm{x}^{2}+5 \mathrm{x}-36} \cdot \frac{?}{?}=\frac{\mathrm{x}(\mathrm{x}+4)}{4(\mathrm{x}+9)}$

Accepted Answer

A)  $\frac{x^{2}+16}{4 x}$ 
B)  $\frac{x(x-4)}{4(x-9)}$ 
C)  $\frac{x+4}{4 x(x-9)}$ 
D)  $\frac{x^{2}-16}{4 x}$ 
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 16

Evaluate the rational function.
-$r(x)=\frac{x}{x^{2}-9 x+20}, r(7)$

Accepted Answer

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

Question 17

Find all values of the variable for which the rational expression is undefined.
-$\frac{x^{2}-36}{x^{2}-17 x+72}$

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

A)  $\frac{4 x^{2}-21 x+18}{x^{2}-12 x+36}$ 
B)  $\frac{4 \mathrm{x}+3}{(\mathrm{x}+6)(\mathrm{x}-6)}$ 
C)  $\frac{4 x^{2}-21 x+18}{x^{2}-36}$ 
D)  $\frac{4 x^{2}-21 x+18}{x^{2}+12 x+36}$ 
A)  $\frac{4 x^{2}-21 x+18}{x^{2}-12 x+36}$ 
B)  $\frac{4 \mathrm{x}+3}{(\mathrm{x}+6)(\mathrm{x}-6)}$ 
C)  $\frac{4 x^{2}-21 x+18}{x^{2}-36}$ 
D)  $\frac{4 x^{2}-21 x+18}{x^{2}+12 x+36}$

Question 20

$\frac{1-7 \mathrm{x}}{\mathrm{x}^{2}-10 \mathrm{x}+9}+\frac{8-2 \mathrm{x}}{\mathrm{x}^{2}-10 \mathrm{x}+9}$

Accepted Answer

A)  $-\frac{9}{9-x}$ 
B)  $-\frac{9}{x+9}$ 
C)  $-\frac{9}{x-9}$ 
D)  $\frac{9}{x-9}$ 
A)  $-\frac{9}{9-x}$ 
B)  $-\frac{9}{x+9}$ 
C)  $-\frac{9}{x-9}$ 
D)  $\frac{9}{x-9}$

Find the missing numerator and denominator. - $\frac{x^{2}-1}{x^{2}+6 x+9} \div \frac{?}{?}=\frac{(x+1)(x-5)}{5(x+3)}$

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

For the given rational functions $f(x)$ and $g(x), f(x)+g(x)$ . - $f(x)=\frac{x}{16-x^{2}}, g(x)=\frac{3}{8 x-32}$

$\frac{3 x}{x-4}-\frac{12}{x-4}$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \div g(x)$ . - $f(x)=\frac{x^{2}+8 x+15}{x^{2}+12 x+35}, g(x)=\frac{x^{2}+3 x}{x^{2}-2 x-63}$

$\frac{8 x+4 y}{5 x^{2} y}-\frac{3 x-3 y}{x y^{2}}$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \div g(x)$ . - $f(x)=(x+6), g(x)=\frac{3 x^{2}+12 x-36}{x^{2}+7 x+6}$

A plane flies 400 miles with the wind and 350 miles against the wind in the same length of time. If the speed of the wind is $25 \mathrm{mph}$ , what is the speed of the plane in calm air?

$\frac{\mathrm{x}}{\mathrm{x}^{2}+6 \mathrm{x}-27}+\frac{9}{\mathrm{x}^{2}+6 \mathrm{x}-27}$

One number is one larger than another positive number. If the reciprocal of the smaller number is multiplied by the reciprocal of the larger number, the product is $\frac{1}{30}$ . Find the two numbers.

A number minus 12 times its reciprocal is 1 . Find the number.

$\frac{x}{4}-\frac{x}{7}=5$

$\frac{x^{2}+14 x+13}{x^{2}+5 x+6}+\frac{x^{2}-8 x-9}{x^{2}+5 x+6}$

Find the missing numerator and denominator. - $\frac{\mathrm{x}^{2}}{\mathrm{x}^{2}+5 \mathrm{x}-36} \cdot \frac{?}{?}=\frac{\mathrm{x}(\mathrm{x}+4)}{4(\mathrm{x}+9)}$

Evaluate the rational function. - $r(x)=\frac{x}{x^{2}-9 x+20}, r(7)$

Find all values of the variable for which the rational expression is undefined. - $\frac{x^{2}-36}{x^{2}-17 x+72}$

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

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

$\frac{1-7 \mathrm{x}}{\mathrm{x}^{2}-10 \mathrm{x}+9}+\frac{8-2 \mathrm{x}}{\mathrm{x}^{2}-10 \mathrm{x}+9}$

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

Find the missing numerator and denominator. - x2−1x2+6x+9÷??=(x+1)(x−5)5(x+3)\frac{x^{2}-1}{x^{2}+6 x+9} \div \frac{?}{?}=\frac{(x+1)(x-5)}{5(x+3)}x2+6x+9x2−1​÷??​=5(x+3)(x+1)(x−5)​

x+4x−3⋅x−6x+3\frac{x+4}{x-3} \cdot \frac{x-6}{x+3}x−3x+4​⋅x+3x−6​

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)=x16−x2,g(x)=38x−32f(x)=\frac{x}{16-x^{2}}, g(x)=\frac{3}{8 x-32}f(x)=16−x2x​,g(x)=8x−323​

3xx−4−12x−4\frac{3 x}{x-4}-\frac{12}{x-4}x−43x​−x−412​

For the given functions f(x)f(x)f(x) and g(x)g(x)g(x) , find f(x)÷g(x)f(x) \div g(x)f(x)÷g(x) . - f(x)=x2+8x+15x2+12x+35,g(x)=x2+3xx2−2x−63f(x)=\frac{x^{2}+8 x+15}{x^{2}+12 x+35}, g(x)=\frac{x^{2}+3 x}{x^{2}-2 x-63}f(x)=x2+12x+35x2+8x+15​,g(x)=x2−2x−63x2+3x​

8x+4y5x2y−3x−3yxy2\frac{8 x+4 y}{5 x^{2} y}-\frac{3 x-3 y}{x y^{2}}5x2y8x+4y​−xy23x−3y​

For the given functions f(x)f(x)f(x) and g(x)g(x)g(x) , find f(x)÷g(x)f(x) \div g(x)f(x)÷g(x) . - f(x)=(x+6),g(x)=3x2+12x−36x2+7x+6f(x)=(x+6), g(x)=\frac{3 x^{2}+12 x-36}{x^{2}+7 x+6}f(x)=(x+6),g(x)=x2+7x+63x2+12x−36​

A plane flies 400 miles with the wind and 350 miles against the wind in the same length of time. If the speed of the wind is 25mph25 \mathrm{mph}25mph , what is the speed of the plane in calm air?

xx2+6x−27+9x2+6x−27\frac{\mathrm{x}}{\mathrm{x}^{2}+6 \mathrm{x}-27}+\frac{9}{\mathrm{x}^{2}+6 \mathrm{x}-27}x2+6x−27x​+x2+6x−279​

One number is one larger than another positive number. If the reciprocal of the smaller number is multiplied by the reciprocal of the larger number, the product is 130\frac{1}{30}301​ . Find the two numbers.

A number minus 12 times its reciprocal is 1 . Find the number.

x4−x7=5\frac{x}{4}-\frac{x}{7}=54x​−7x​=5

x2+14x+13x2+5x+6+x2−8x−9x2+5x+6\frac{x^{2}+14 x+13}{x^{2}+5 x+6}+\frac{x^{2}-8 x-9}{x^{2}+5 x+6}x2+5x+6x2+14x+13​+x2+5x+6x2−8x−9​

Find the missing numerator and denominator. - x2x2+5x−36⋅??=x(x+4)4(x+9)\frac{\mathrm{x}^{2}}{\mathrm{x}^{2}+5 \mathrm{x}-36} \cdot \frac{?}{?}=\frac{\mathrm{x}(\mathrm{x}+4)}{4(\mathrm{x}+9)}x2+5x−36x2​⋅??​=4(x+9)x(x+4)​

Evaluate the rational function. - r(x)=xx2−9x+20,r(7)r(x)=\frac{x}{x^{2}-9 x+20}, r(7)r(x)=x2−9x+20x​,r(7)

Find all values of the variable for which the rational expression is undefined. - x2−36x2−17x+72\frac{x^{2}-36}{x^{2}-17 x+72}x2−17x+72x2−36​

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

4xx+6+3x−6\frac{4 \mathrm{x}}{\mathrm{x}+6}+\frac{3}{\mathrm{x}-6}x+64x​+x−63​

1−7xx2−10x+9+8−2xx2−10x+9\frac{1-7 \mathrm{x}}{\mathrm{x}^{2}-10 \mathrm{x}+9}+\frac{8-2 \mathrm{x}}{\mathrm{x}^{2}-10 \mathrm{x}+9}x2−10x+91−7x​+x2−10x+98−2x​

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

Find the missing numerator and denominator. - $\frac{x^{2}-1}{x^{2}+6 x+9} \div \frac{?}{?}=\frac{(x+1)(x-5)}{5(x+3)}$

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

For the given rational functions $f(x)$ and $g(x), f(x)+g(x)$ . - $f(x)=\frac{x}{16-x^{2}}, g(x)=\frac{3}{8 x-32}$

$\frac{3 x}{x-4}-\frac{12}{x-4}$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \div g(x)$ . - $f(x)=\frac{x^{2}+8 x+15}{x^{2}+12 x+35}, g(x)=\frac{x^{2}+3 x}{x^{2}-2 x-63}$

$\frac{8 x+4 y}{5 x^{2} y}-\frac{3 x-3 y}{x y^{2}}$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \div g(x)$ . - $f(x)=(x+6), g(x)=\frac{3 x^{2}+12 x-36}{x^{2}+7 x+6}$

A plane flies 400 miles with the wind and 350 miles against the wind in the same length of time. If the speed of the wind is $25 \mathrm{mph}$ , what is the speed of the plane in calm air?

$\frac{\mathrm{x}}{\mathrm{x}^{2}+6 \mathrm{x}-27}+\frac{9}{\mathrm{x}^{2}+6 \mathrm{x}-27}$

One number is one larger than another positive number. If the reciprocal of the smaller number is multiplied by the reciprocal of the larger number, the product is $\frac{1}{30}$ . Find the two numbers.

$\frac{x}{4}-\frac{x}{7}=5$

$\frac{x^{2}+14 x+13}{x^{2}+5 x+6}+\frac{x^{2}-8 x-9}{x^{2}+5 x+6}$

Find the missing numerator and denominator. - $\frac{\mathrm{x}^{2}}{\mathrm{x}^{2}+5 \mathrm{x}-36} \cdot \frac{?}{?}=\frac{\mathrm{x}(\mathrm{x}+4)}{4(\mathrm{x}+9)}$

Evaluate the rational function. - $r(x)=\frac{x}{x^{2}-9 x+20}, r(7)$

Find all values of the variable for which the rational expression is undefined. - $\frac{x^{2}-36}{x^{2}-17 x+72}$

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

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

$\frac{1-7 \mathrm{x}}{\mathrm{x}^{2}-10 \mathrm{x}+9}+\frac{8-2 \mathrm{x}}{\mathrm{x}^{2}-10 \mathrm{x}+9}$