Exam 7: Rational Expressions and Functions

Choose the one alternative that best completes the statement or answers the question. Without actually solving the equation, list all possible numbers that would have to be rejected if they appeared as potential solutions. - $\frac{9}{x^{2}-5 x}+\frac{1}{x+6}=\frac{6}{x-8}$

(Multiple Choice)

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Question 61

Express the rational expression in lowest terms. - $\frac{y^{2}+9 y+20}{y^{2}+12 y+32}$

(Multiple Choice)

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Question 62

Are the two expressions equivalent? Explain your answer. $\frac{p}{p-5}-\frac{7 p+3}{p-5}$ and $\frac{-6 p+3}{p-5}$

(Essay)

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Question 63

Solve the formula for the specified variable. - $\mathrm{A}=\frac{1}{2} \mathrm{~h}(\mathrm{~B}+\mathrm{b})$ for $\mathrm{B}$

(Multiple Choice)

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Question 64

Assume that the expressions given are denominators of fractions. Find the least common denominator. - $7 a+21, a^{2}+3 a$

(Multiple Choice)

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Question 65

Find all numbers that are not in the domain of the function. Then give the domain using set notation. - $f(x)=\frac{x^{2}-10 x}{2 x}$

(Multiple Choice)

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Question 66

Perform the indicated operation and express in lowest terms. - $\frac{4 x^{2}-19 x y-5 y^{2}}{y^{2}+4 x y-5 x^{2}} \cdot \frac{y^{2}+3 x y-4 x^{2}}{8 x^{2}+6 x y+y^{2}} \div \frac{4 x^{2}-19 x y-5 y^{2}}{10 x^{2}+3 x y-y^{2}}$

(Multiple Choice)

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Question 67

Simplify the complex fraction. - $\frac{a^{-2}-b^{-1}}{7 a^{-1}+3 b^{-2}}$

(Multiple Choice)

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Question 68

Tell whether or not the rational expressions are equivalent. - $\frac{a-b}{a^{2}-b^{2}}, \frac{1}{a+b}$

(True/False)

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Question 69

Write the rational expression in lowest terms. - $\frac{3 \mathrm{k}-15}{10-2 \mathrm{k}}$

(Multiple Choice)

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Question 70

Determine whether the equation represents direct, inverse, joint, or combined variation. - $\mathrm{A}=\pi \mathrm{r}^{2}$ , where $\mathrm{A}$ is the area of a circle of radius $\mathrm{r}$ .

(Short Answer)

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Question 71

$\frac{3}{x-6}+\frac{4}{x}=\frac{-24}{x^{2}-6 x}$

(Multiple Choice)

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Question 72

Determine whether the equation represents direct, inverse, joint, or combined variation. - $r=d / t$ , where $r$ is the rate by which distance $d$ is covered in time $t$

(Short Answer)

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Question 73

Choose the one alternative that best completes the statement or answers the question. -If $\mathrm{f}$ varies jointly as $\mathrm{q}^{2}$ and $\mathrm{h}$ , and $\mathrm{f}=81$ when $\mathrm{q}=3$ and $\mathrm{h}=3$ , find $\mathrm{f}$ when $\mathrm{q}=4$ and $\mathrm{h}=3$ .

(Multiple Choice)

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Question 74

Frank can type a report in 2 hours and James takes 6 hours. How long will it take the two of them typing together?

(Multiple Choice)

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Question 75

Choose the one alternative that best completes the statement or answers the question. Without actually solving the equation, list all possible numbers that would have to be rejected if they appeared as potential solutions. - $\frac{8}{6 x+13}-\frac{1}{x}=\frac{1}{13 x-9}$

(Multiple Choice)

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Question 76

Express the rational expression in lowest terms. - $\frac{5 x-5 y-b x+b y}{5 x-5 y+b x-b y}$

(Multiple Choice)

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Question 77

Simplify the expression and solve the equation. -One of the following is an expression to be simplified by algebraic operations, and the other is an equation to be solved. Identify each, and then simplify the one that is an expression and solve the one that is an equation. (a) $\frac{2 x}{7}+\frac{x}{4}=\frac{5}{2}$ (b) $\frac{2 x}{7}+\frac{x}{4}-\frac{5}{2}$

(Multiple Choice)

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Question 78

Use the example to give a step-by-step explanation of the method to add or subtract two fractions with a common denominator. $\frac{15}{p+3}+\frac{5 p}{p+3}$

(Essay)

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Question 79

The speed of a stream is $4 \mathrm{mph}$ . If a boat travels 84 miles downstream in the same time that it takes to travel 42 miles upstream, what is the speed of the boat in still water?

(Multiple Choice)

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Question 80

Choose the one alternative that best completes the statement or answers the question. Without actually solving the equation, list all possible numbers that would have to be rejected if they appeared as potential solutions. - $\frac{9}{x^{2}-5 x}+\frac{1}{x+6}=\frac{6}{x-8}$

Express the rational expression in lowest terms. - $\frac{y^{2}+9 y+20}{y^{2}+12 y+32}$

Are the two expressions equivalent? Explain your answer. $\frac{p}{p-5}-\frac{7 p+3}{p-5}$ and $\frac{-6 p+3}{p-5}$

Solve the formula for the specified variable. - $\mathrm{A}=\frac{1}{2} \mathrm{~h}(\mathrm{~B}+\mathrm{b})$ for $\mathrm{B}$

Assume that the expressions given are denominators of fractions. Find the least common denominator. - $7 a+21, a^{2}+3 a$

Find all numbers that are not in the domain of the function. Then give the domain using set notation. - $f(x)=\frac{x^{2}-10 x}{2 x}$

Perform the indicated operation and express in lowest terms. - $\frac{4 x^{2}-19 x y-5 y^{2}}{y^{2}+4 x y-5 x^{2}} \cdot \frac{y^{2}+3 x y-4 x^{2}}{8 x^{2}+6 x y+y^{2}} \div \frac{4 x^{2}-19 x y-5 y^{2}}{10 x^{2}+3 x y-y^{2}}$

Simplify the complex fraction. - $\frac{a^{-2}-b^{-1}}{7 a^{-1}+3 b^{-2}}$

Tell whether or not the rational expressions are equivalent. - $\frac{a-b}{a^{2}-b^{2}}, \frac{1}{a+b}$

Write the rational expression in lowest terms. - $\frac{3 \mathrm{k}-15}{10-2 \mathrm{k}}$

Determine whether the equation represents direct, inverse, joint, or combined variation. - $\mathrm{A}=\pi \mathrm{r}^{2}$ , where $\mathrm{A}$ is the area of a circle of radius $\mathrm{r}$ .

$\frac{3}{x-6}+\frac{4}{x}=\frac{-24}{x^{2}-6 x}$

Determine whether the equation represents direct, inverse, joint, or combined variation. - $r=d / t$ , where $r$ is the rate by which distance $d$ is covered in time $t$

Choose the one alternative that best completes the statement or answers the question. -If $\mathrm{f}$ varies jointly as $\mathrm{q}^{2}$ and $\mathrm{h}$ , and $\mathrm{f}=81$ when $\mathrm{q}=3$ and $\mathrm{h}=3$ , find $\mathrm{f}$ when $\mathrm{q}=4$ and $\mathrm{h}=3$ .

Frank can type a report in 2 hours and James takes 6 hours. How long will it take the two of them typing together?

Choose the one alternative that best completes the statement or answers the question. Without actually solving the equation, list all possible numbers that would have to be rejected if they appeared as potential solutions. - $\frac{8}{6 x+13}-\frac{1}{x}=\frac{1}{13 x-9}$

Express the rational expression in lowest terms. - $\frac{5 x-5 y-b x+b y}{5 x-5 y+b x-b y}$

Use the example to give a step-by-step explanation of the method to add or subtract two fractions with a common denominator. $\frac{15}{p+3}+\frac{5 p}{p+3}$

The speed of a stream is $4 \mathrm{mph}$ . If a boat travels 84 miles downstream in the same time that it takes to travel 42 miles upstream, what is the speed of the boat in still water?

Review of the Real Number System

Linear Equations, Inequalities, and Applications

Linear Equations, Graphs, and Functions

Systems of Linear Equations

Exponents, Polynomials, and Polynomial Functions

Factoring

Roots, Radicals, and Root Functions

Quadratic Equations, Inequalities, and Functions

Inverse, Exponential, and Logarithmic Functions

Nonlinear Functions, Conic Sections, and Nonlinear Systems

Further Topics in Algebra

Appendices

Filters

Exam 7: Rational Expressions and Functions

Express the rational expression in lowest terms. - y2+9y+20y2+12y+32\frac{y^{2}+9 y+20}{y^{2}+12 y+32}y2+12y+32y2+9y+20​

Are the two expressions equivalent? Explain your answer. pp−5−7p+3p−5\frac{p}{p-5}-\frac{7 p+3}{p-5}p−5p​−p−57p+3​ and −6p+3p−5\frac{-6 p+3}{p-5}p−5−6p+3​

Solve the formula for the specified variable. - A=12 h( B+b)\mathrm{A}=\frac{1}{2} \mathrm{~h}(\mathrm{~B}+\mathrm{b})A=21​ h( B+b) for B\mathrm{B}B

Assume that the expressions given are denominators of fractions. Find the least common denominator. - 7a+21,a2+3a7 a+21, a^{2}+3 a7a+21,a2+3a

Find all numbers that are not in the domain of the function. Then give the domain using set notation. - f(x)=x2−10x2xf(x)=\frac{x^{2}-10 x}{2 x}f(x)=2xx2−10x​

Simplify the complex fraction. - a−2−b−17a−1+3b−2\frac{a^{-2}-b^{-1}}{7 a^{-1}+3 b^{-2}}7a−1+3b−2a−2−b−1​

Tell whether or not the rational expressions are equivalent. - a−ba2−b2,1a+b\frac{a-b}{a^{2}-b^{2}}, \frac{1}{a+b}a2−b2a−b​,a+b1​

Write the rational expression in lowest terms. - 3k−1510−2k\frac{3 \mathrm{k}-15}{10-2 \mathrm{k}}10−2k3k−15​

Determine whether the equation represents direct, inverse, joint, or combined variation. - A=πr2\mathrm{A}=\pi \mathrm{r}^{2}A=πr2 , where A\mathrm{A}A is the area of a circle of radius r\mathrm{r}r .

3x−6+4x=−24x2−6x\frac{3}{x-6}+\frac{4}{x}=\frac{-24}{x^{2}-6 x}x−63​+x4​=x2−6x−24​

Determine whether the equation represents direct, inverse, joint, or combined variation. - r=d/tr=d / tr=d/t , where rrr is the rate by which distance ddd is covered in time ttt

Frank can type a report in 2 hours and James takes 6 hours. How long will it take the two of them typing together?

Express the rational expression in lowest terms. - 5x−5y−bx+by5x−5y+bx−by\frac{5 x-5 y-b x+b y}{5 x-5 y+b x-b y}5x−5y+bx−by5x−5y−bx+by​

Use the example to give a step-by-step explanation of the method to add or subtract two fractions with a common denominator. 15p+3+5pp+3\frac{15}{p+3}+\frac{5 p}{p+3}p+315​+p+35p​

The speed of a stream is 4mph4 \mathrm{mph}4mph . If a boat travels 84 miles downstream in the same time that it takes to travel 42 miles upstream, what is the speed of the boat in still water?

Review of the Real Number System

Linear Equations, Inequalities, and Applications

Linear Equations, Graphs, and Functions

Systems of Linear Equations

Exponents, Polynomials, and Polynomial Functions

Factoring

Roots, Radicals, and Root Functions

Quadratic Equations, Inequalities, and Functions

Inverse, Exponential, and Logarithmic Functions

Nonlinear Functions, Conic Sections, and Nonlinear Systems

Further Topics in Algebra

Appendices

Filters

Express the rational expression in lowest terms. - $\frac{y^{2}+9 y+20}{y^{2}+12 y+32}$

Are the two expressions equivalent? Explain your answer. $\frac{p}{p-5}-\frac{7 p+3}{p-5}$ and $\frac{-6 p+3}{p-5}$

Solve the formula for the specified variable. - $\mathrm{A}=\frac{1}{2} \mathrm{~h}(\mathrm{~B}+\mathrm{b})$ for $\mathrm{B}$

Assume that the expressions given are denominators of fractions. Find the least common denominator. - $7 a+21, a^{2}+3 a$

Find all numbers that are not in the domain of the function. Then give the domain using set notation. - $f(x)=\frac{x^{2}-10 x}{2 x}$

Simplify the complex fraction. - $\frac{a^{-2}-b^{-1}}{7 a^{-1}+3 b^{-2}}$

Tell whether or not the rational expressions are equivalent. - $\frac{a-b}{a^{2}-b^{2}}, \frac{1}{a+b}$

Write the rational expression in lowest terms. - $\frac{3 \mathrm{k}-15}{10-2 \mathrm{k}}$

Determine whether the equation represents direct, inverse, joint, or combined variation. - $\mathrm{A}=\pi \mathrm{r}^{2}$ , where $\mathrm{A}$ is the area of a circle of radius $\mathrm{r}$ .

$\frac{3}{x-6}+\frac{4}{x}=\frac{-24}{x^{2}-6 x}$

Determine whether the equation represents direct, inverse, joint, or combined variation. - $r=d / t$ , where $r$ is the rate by which distance $d$ is covered in time $t$

Express the rational expression in lowest terms. - $\frac{5 x-5 y-b x+b y}{5 x-5 y+b x-b y}$

Use the example to give a step-by-step explanation of the method to add or subtract two fractions with a common denominator. $\frac{15}{p+3}+\frac{5 p}{p+3}$

The speed of a stream is $4 \mathrm{mph}$ . If a boat travels 84 miles downstream in the same time that it takes to travel 42 miles upstream, what is the speed of the boat in still water?