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Intermediate Algebra 2e

Review Exercises

Intermediate Algebra 2eReview Exercises
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Review Exercises

Simplify, Multiply, and Divide Rational Expressions

Determine the Values for Which a Rational Expression is Undefined

In the following exercises, determine the values for which the rational expression is undefined.

377.

5a+33a25a+33a2

378.

b7b225b7b225

379.

5x2y28y5x2y28y

380.

x3x2x30x3x2x30

Simplify Rational Expressions

In the following exercises, simplify.

381.

18241824

382.

9m418mn39m418mn3

383.

x2+7x+12x2+8x+16x2+7x+12x2+8x+16

384.

7v3525v27v3525v2

Multiply Rational Expressions

In the following exercises, multiply.

385.

58·41558·415

386.

3xy28y3·16y224x3xy28y3·16y224x

387.

72x12x28x+32·x2+10x+24x23672x12x28x+32·x2+10x+24x236

388.

2y2+y34y2·y24y+42y2+11y+122y2+y34y2·y24y+42y2+11y+12

Divide Rational Expressions

In the following exercises, divide.

389.

x24x12x2+8x+12÷x2363xx24x12x2+8x+12÷x2363x

390.

y2164÷y3642y2+8y+32y2164÷y3642y2+8y+32

391.

11+ww9÷121w29w11+ww9÷121w29w

392.

3y212y634y+3÷(6y242y)3y212y634y+3÷(6y242y)

393.

c2643c2+26c+16c24c3215c+10c2643c2+26c+16c24c3215c+10

394.

8a2+16aa4·a2+2a24a2+7a+10÷2a26aa+58a2+16aa4·a2+2a24a2+7a+10÷2a26aa+5

Multiply and Divide Rational Functions

395.

Find R(x)=f(x)·g(x)R(x)=f(x)·g(x) where f(x)=9x2+9xx23x4f(x)=9x2+9xx23x4 and g(x)=x2163x2+12x.g(x)=x2163x2+12x.

396.

Find R(x)=f(x)g(x)R(x)=f(x)g(x) where f(x)=27x23x21f(x)=27x23x21 and
g(x)=9x2+54xx2x42.g(x)=9x2+54xx2x42.

Add and Subtract Rational Expressions

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, perform the indicated operations.

397.

715+815715+815

398.

4a22a112a14a22a112a1

399.

y2+10yy+5+25y+5y2+10yy+5+25y+5

400.

7x2x29+21xx297x2x29+21xx29

401.

x2x73x+28x7x2x73x+28x7

402.

y2y+11121y+11y2y+11121y+11

403.

4q2q+3q2+6q+53q2q6q2+6q+54q2q+3q2+6q+53q2q6q2+6q+5

404.

5t2+4t+3t2254t28t32t2255t2+4t+3t2254t28t32t225

Add and Subtract Rational Expressions Whose Denominators Are Opposites

In the following exercises, add and subtract.

405.

18w6w1+3w216w18w6w1+3w216w

406.

a2+3aa243a+84a2a2+3aa243a+84a2

407.

2b2+3b15b249b2+16b149b22b2+3b15b249b2+16b149b2

408.

8y210y+72y5+2y2+7y+252y8y210y+72y5+2y2+7y+252y

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

409.

7a23a10,3aa2a207a23a10,3aa2a20

410.

6n24,2nn24n+46n24,2nn24n+4

411.

53p2+17p6,2m3p2+23p853p2+17p6,2m3p2+23p8

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

412.

75a+32b75a+32b

413.

2c2+9c+32c2+9c+3

414.

3xx29+5x2+6x+93xx29+5x2+6x+9

415.

2xx2+10x+24+3xx2+8x+162xx2+10x+24+3xx2+8x+16

416.

5qp2qp2+4qq215qp2qp2+4qq21

417.

3yy+2y+2y+83yy+2y+2y+8

418.

−3w15w2+w20w+24w−3w15w2+w20w+24w

419.

7m+3m+257m+3m+25

420.

nn+3+2n3n+9n29nn+3+2n3n+9n29

421.

8aa2644a+88aa2644a+8

422.

512x2y+720xy3512x2y+720xy3

Add and Subtract Rational Functions

In the following exercises, find R(x)=f(x)+g(x)R(x)=f(x)+g(x) where f(x)f(x) and g(x)g(x) are given.

423.

f(x)=2x2+12x11x2+3x10,f(x)=2x2+12x11x2+3x10,g(x)=x+12xg(x)=x+12x

424.

f(x)=−4x+31x2+x30,f(x)=−4x+31x2+x30,g(x)=5x+6g(x)=5x+6

In the following exercises, find R(x)=f(x)g(x)R(x)=f(x)g(x) where f(x)f(x) and g(x)g(x) are given.

425.

f(x)=4xx2121,f(x)=4xx2121,g(x)=2x11g(x)=2x11

426.

f(x)=7x+6,f(x)=7x+6,g(x)=14xx236g(x)=14xx236

Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

427.

7xx+214x2x247xx+214x2x24

428.

25+5613+1425+5613+14

429.

x3xx+51x+5+1x5x3xx+51x+5+1x5

430.

2m+mnnm1n2m+mnnm1n

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

431.

13+1814+11213+1814+112

432.

3a21b1a+1b23a21b1a+1b2

433.

2z249+1z+79z+7+12z72z249+1z+79z+7+12z7

434.

3y24y322y8+1y+43y24y322y8+1y+4

7.4 Solve Rational Equations

Solve Rational Equations

In the following exercises, solve.

435.

12+23=1x12+23=1x

436.

12m=8m212m=8m2

437.

1b2+1b+2=3b241b2+1b+2=3b24

438.

3q+82q2=13q+82q2=1

439.

v15v29v+18=4v3+2v6v15v29v+18=4v3+2v6

440.

z12+z+33z=1zz12+z+33z=1z

Solve Rational Equations that Involve Functions

441.

For rational function, f(x)=x+2x26x+8,f(x)=x+2x26x+8, find the domain of the function solve f(x)=1f(x)=1 find the points on the graph at this function value.

442.

For rational function, f(x)=2xx2+7x+10,f(x)=2xx2+7x+10, find the domain of the function solve f(x)=2f(x)=2 find the points on the graph at this function value.

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

443.

Vl=hwVl=hw for l.l.

444.

1x2y=51x2y=5 for y.y.

445.

x=y+5z7x=y+5z7 for z.z.

446.

P=kVP=kV for V.V.

Solve Applications with Rational Equations

Solve Proportions

In the following exercises, solve.

447.

x4=35x4=35

448.

3y=953y=95

449.

ss+20=37ss+20=37

450.

t35=t+29t35=t+29

Solve Using Proportions

In the following exercises, solve.

451.

Rachael had a 21-ounce strawberry shake that has 739 calories. How many calories are there in a 32-ounce shake?

452.

Leo went to Mexico over Christmas break and changed $525 dollars into Mexican pesos. At that time, the exchange rate had $1 US is equal to 16.25 Mexican pesos. How many Mexican pesos did he get for his trip?

Solve Similar Figure Applications

In the following exercises, solve.

453.

ΔABCΔABC is similar to ΔXYZ.ΔXYZ. The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.

The first figure is triangle A B C with side A B 8 units long, side B C 7 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 2 and two-thirds units long, side Y Z x units long, and side X Z 3 units long.
454.

On a map of Europe, Paris, Rome, and Vienna form a triangle whose sides are shown in the figure below. If the actual distance from Rome to Vienna is 700 miles, find the distance from
Paris to Rome
Paris to Vienna

The figure is a triangle formed by Paris, Vienna, and Rome. The distance between Paris and Vienna is 7.7 centimeters. The distance between Vienna and Rome is 7 centimeters. The distance between Rome and Paris is 8.9 centimeters.
455.

Francesca is 5.75 feet tall. Late one afternoon, her shadow was 8 feet long. At the same time, the shadow of a nearby tree was 32 feet long. Find the height of the tree.

456.

The height of a lighthouse in Pensacola, Florida is 150 feet. Standing next to the statue, 5.5-foot-tall Natasha cast a 1.1-foot shadow. How long would the shadow of the lighthouse be?

Solve Uniform Motion Applications

In the following exercises, solve.

457.

When making the 5-hour drive home from visiting her parents, Lolo ran into bad weather. She was able to drive 176 miles while the weather was good, but then driving 10 mph slower, went 81 miles when it turned bad. How fast did she drive when the weather was bad?

458.

Mark is riding on a plane that can fly 490 miles with a tailwind of 20 mph in the same time that it can fly 350 miles against a tailwind of 20 mph. What is the speed of the plane?

459.

Josue can ride his bicycle 8 mph faster than Arjun can ride his bike. It takes Arjun 3 hours longer than Josue to ride 48 miles. How fast can Josue ride his bike?

460.

Curtis was training for a triathlon. He ran 8 kilometers and biked 32 kilometers in a total of 3 hours. His running speed was 8 kilometers per hour less than his biking speed. What was his running speed?

Solve Work Applications

In the following exercises, solve.

461.

Brandy can frame a room in 1 hour, while Jake takes 4 hours. How long could they frame a room working together?

462.

Prem takes 3 hours to mow the lawn while her cousin, Barb, takes 2 hours. How long will it take them working together?

463.

Jeffrey can paint a house in 6 days, but if he gets a helper he can do it in 4 days. How long would it take the helper to paint the house alone?

464.

Marta and Deb work together writing a book that takes them 90 days. If Marta worked alone it would take her 120 days. How long would it take Deb to write the book alone?

Solve Direct Variation Problems

In the following exercises, solve.

465.

If yy varies directly as xx when y=9y=9 and x=3,x=3, find xx when y=21.y=21.

466.

If yy varies inversely as xx when y=20y=20 and x=2,x=2, find yy when x=4.x=4.

467.

Vanessa is traveling to see her fiancé. The distance, d,d, varies directly with the speed, v,v, she drives. If she travels 258 miles driving 60 mph, how far would she travel going 70 mph?

468.

If the cost of a pizza varies directly with its diameter, and if an 8” diameter pizza costs $12, how much would a 6” diameter pizza cost?

469.

The distance to stop a car varies directly with the square of its speed. It takes 200 feet to stop a car going 50 mph. How many feet would it take to stop a car going 60 mph?

Solve Inverse Variation Problems

In the following exercises, solve.

470.

If mm varies inversely with the square of n,n, when m=4m=4 and n=6n=6 find mm when n=2.n=2.

471.

The number of tickets for a music fundraiser varies inversely with the price of the tickets. If Madelyn has just enough money to purchase 12 tickets for $6, how many tickets can Madelyn afford to buy if the price increased to $8?

472.

On a string instrument, the length of a string varies inversely with the frequency of its vibrations. If an 11-inch string on a violin has a frequency of 360 cycles per second, what frequency does a 12-inch string have?

Solve Rational Inequalities

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

473.

x3x+40x3x+40

474.

5xx2>15xx2>1

475.

3x2x423x2x42

476.

1x24x12<01x24x12<0

477.

124x21x124x21x

478.

4x2<3x+14x2<3x+1

Solve an Inequality with Rational Functions

In the following exercises, solve each rational function inequality and write the solution in interval notation

479.

Given the function, R(x)=x5x2,R(x)=x5x2, find the values of xx that make the function greater than or equal to 0.

480.

Given the function, R(x)=x+1x+3,R(x)=x+1x+3, find the values of xx that make the function less than or equal to 0.

481.

The function
C(x)=150x+100,000C(x)=150x+100,000 represents the cost to produce x,x, number of items. Find the average cost function, c(x)c(x) how many items should be produced so that the average cost is less than $160.

482.

Tillman is starting his own business by selling tacos at the beach. Accounting for the cost of his food truck and ingredients for the tacos, the function C(x)=2x+6,000C(x)=2x+6,000 represents the cost for Tillman to produce x,x, tacos. Find the average cost function, c(x)c(x) for Tillman’s Tacos how many tacos should Tillman produce so that the average cost is less than $4.

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