1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

4. Polynomial and Rational Functions

5. Exponential and Logarithmic Functions

6. Systems of Equations and Inequalities

7. Matrices and Determinants

8. Conic Sections

9. Sequences, Induction, and Probability

undefined - Channels

Simplifying Complex Fractions

In Exercises 47–58, multiply as indicated.

2(-3)(-1)(-2)(-4)

Find the domain of each rational expression. 
(2x - 4) / (x + 7)

In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s).

4+5(x - 7)³, for x=9

In Exercises 51–72, multiply as indicated.

(x−3) ⋅ x²+x+1/x²−5x+6

Simplifying Complex Fractions - Ex 3

In Exercises 71–82, divide as indicated or state that the division is undefined.

6 ÷ (- 2/5)

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

A number increased by six

In Exercises 83–100, use the order of operations to simplify each expression.

15-√3-(-1) + 12÷2•3

Adding and Subtracting Rational Expressions

Simplify each complex fraction. [1 + (1/x)] / (1-1/x)

Rational Expressions: Writing in Lowest Terms - Ex 2

In Exercises 47–58, multiply as indicated.

-√2 • 0

Simplify each complex rational expression. (1/x-1/2)/(1/3-x/6)

In Exercises 51–72, multiply as indicated.

x²−49/x²−4x−21 ⋅ x+3/x

Simplifying Complex Fractions - Ex 1

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

Four times a number

Simplify each complex fraction. [ 2 + 2/(1+x) ] / [ 2 - 2/(1-x) ]

In Exercises 51–72, multiply as indicated.

x²+4x+4/x²+8x+16 ⋅ (x+4)³/(x+2)³

Rational Expressions:  Adding and Subtracting.  Ex 1

In Exercises 35–41, simplify each expression.

(-2)⁴+(-3)² / 2²-(-21)

❖ Finding all the Zeros of a Polynomial - Example 3 ❖

Simplify each complex fraction. [ 1/(x^3-y^3) ] / [ 1/(x^2 -y^2) ]

Rational Expressions:  Multiplying and Dividing.  Ex 1

Simplify each rational expression. Also, list all numbers that must be excluded from the domain. [x^3+2x^2]/[x+2]

In Exercises 1–3, write each English phrase as an algebraic expression. Let x represent the number.

Ten less than twice a number

Evaluate each expression for x = -4 and y = 2. |-3x + 4y|

Perform the indicated operation, and write each answer in lowest terms. 2x/5 * 10/x^2

In Exercises 111–120, use the order of operations to simplify each expression. 10^2−100÷5^2⋅2−3

In Exercises 27–50, simplify each rational expression. If the rational expression cannot be simplified, so state.

(x³+4x²−3x−12)/(x+4)

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). 4+5(x-7)^3, for x =9

Simplify each complex rational expression. [3-1/(x+3)]/[3+1/(x+3)]

In Exercises 27–34, use the roster method to list the elements in each set.

{x | x is an odd whole number less than 11}

Simplify each complex fraction. (2 - 2/y) / (2 + 2/y)

In Exercises 83–100, use the order of operations to simplify each expression.

[5•2-3²]/[3²-(-2)]²

In Exercises 71–82, divide as indicated or state that the division is undefined.

0 / -4.6

Simplify each complex fraction. [ (y+3)/y - 4/(y-1) ] / [ y/(y - 1) + 1/y ]

Simplify each rational expression. Assume all variable expressions represent positive
real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [2(2x-3)^1/3 - (x-1)(2x-3)^-2/3] / [(2x-2)^-2/3]

In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s).

x²+3x, for x=1/3

In Exercises 83–100, use the order of operations to simplify each expression.

8-3[-2(2-5)-4(8-6)]

Write each rational expression in lowest terms. x^3 + 64 / x + 4

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

Three less than the quotient of five and a number

In Exercises 35–41, simplify each expression.

-3[4-(6-8)]

Write each rational expression in lowest terms. 3(3 - t) / (t + 5)(t - 3)

Find the domain of each rational expression. (3x + 7) / (4x + 2)(x - 12)

Evaluate each expression for x = -4 and y = 2. 4|x| + 4|y| / |x|

In Exercises 73–90, divide as indicated.

x²−2x/15 ÷ x-2/5

Rewrite each expression using the distributive property and simplify, if possible. 15x-10x

In Exercises 83–100, use the order of operations to simplify each expression.

8² - 16 ÷ 2² • 4 - 3

Multiply or divide, as indicated. (y^3 + y^2)/7 * 49/(y^4 + y^3)

Add or subtract, as indicated. (7x + 8)/(3x + 2) - (x + 4)/(3x + 2)

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

Twice a number

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

Ten more than twice a number

In Exercises 7–8, use the roster method to list the elements in each set.

{x|x is a natural number less than 3}

In Exercises 29–34, find -x for the given value of x.

x=-5

In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s).

6 + 5(x - 6)³, for x=8

In Exercises 97–102, write each algebraic expression without parentheses. 1/3(3x)+[(4y)+(−4y)]

In Exercises 71–82, divide as indicated or state that the division is undefined.

- 4.6/0

Perform the indicated operation, and write each answer in lowest terms. 
2x/5 + x/4

Simplify: 9(10x-4) - (5x-10). (Section 1.2, Example 14)

In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s).

x²-4(x - y), for x=8 and y=3

In Exercises 71–82, divide as indicated or state that the division is undefined.

-90 / -2

Finding the Domain of an Expression Involving Fractions - Example 3

How to Find Restrictions on Rational Expressions

Algebra - Complex Fractions

Multiply or divide, as indicated. (x^2 - y^2)/(x - y)^2 * (x^2 - xy + y^2)/(x^2 - 2xy + y^2) ÷ (x^3 + y^3)/(x - y)^4

Multiply or divide as indicated. [(x^2+6x+9)(x+3)]/[(x^2-4)(x-2)]

In Exercises 111–120, use the order of operations to simplify each expression. 6(−4)−5(−3)/(9−10)

In Exercises 73–90, divide as indicated.

x+5/7 ÷ 4x+20/9

In Exercises 35–46, subtract as indicated.

-20-(-5)

Subtracting rational expressions

Simplify each expression. -1/4(20m+8y-32ɀ)

In Exercises 35–48, use the meaning of the symbols ∈ and ∉ to determine whether each statement is true or false.

7 ∈ {x | x is an integer}

Multiply or divide, as indicated. (2k + 8)/6 ÷ (3k + 12)/2

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). x^2-6x+3, for x=7

In Exercises 85–96, simplify each algebraic expression. 5(3y−2)−(7y+2)

In Exercises 83–100, use the order of operations to simplify each expression.

[2(-2)-4(-3)]/(5-8)

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). 6x-y, for x=3 and y=8

Rational Expressions:  Multiplying and Dividing.  Ex 3

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

Four less than a number

In Exercises 22–33, perform the indicated operation.

-7-(-20)

In Exercises 35–46, subtract as indicated.

0-(-√2)

In Exercises 1–14, simplify the expression or solve the equation, whichever is appropriate.

-4²÷2+(-3)(-5)

In Exercises 111–116, use the distributive property to write an equivalent expression..

-(3x-6)

In Exercises 35–46, subtract as indicated.

8-(-10)

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). x^2+3x, for x=8

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). x^2-3(x-y), for x=8 and y=2

In Exercises 9–11, determine whether each statement is true or false.

0 ∈ {x|x is a natural number}

In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s).

8+6x, for x=5

In Exercises 85–96, simplify each algebraic expression. 18x^2+4−[6(x^2−2)+5]

Simplify each rational expression. Assume all variable expressions represent positive
real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [(x^2 +1)^4(2x) - x^2(4)(x^2+1)^3(2x)] / [(x^2+1)^8]

In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s).

x²-6x+3, for x=7

In Exercises 111–116, use the distributive property to write an equivalent expression..

3(2x+5)

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

The quotient of six and the difference of ten and a number

In Exercises 1–14, simplify the expression or solve the equation, whichever is appropriate.

-2⁴+(-2)² / -4-(2-2)

In Exercises 71–82, divide as indicated or state that the division is undefined.

- 1/2 ÷ (- 7/9)

Solve: 4x-1 / 10 = 5x+2 / 4 - 4. (Section 1.4, Example 4)

In Exercises 83–100, use the order of operations to simplify each expression.

[(5-6)²-2|3-7|]/(89-3•5²)

Find the domain of each rational expression. (x + 3) / (x - 6)

Dividing rational expressions

Rational Expressions:  Adding and Subtracting.  Ex 2

In Exercises 73–90, divide as indicated.

y²−25/2y−2 ÷ y²+10y+25/y²+4y−5

In Exercises 101–104, write an algebraic expression equivalent to the given expression using each of the commutative properties.

7x-5

Add or subtract, as indicated. 3/(a - 2) - 1/(2 - a)

In Exercises 33–68, add or subtract as indicated. 4/(x^2+6x+9) + 4/(x+3)

In Exercises 69–82, simplify each complex rational expression. (1/(x+h)^2 − 1/x^2)/h

Find the domain of each rational expression. 12/ (x2 + 5x + 6)

Rational Expressions:  Multiplying and Dividing.  Ex 2

Multiply  Rational Expressions

Simplify each rational expression. Assume all variable expressions represent positive
real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [4(x^2- 1)^3 + 8x(x^2-1)^4] / [16(x^2-1)^3]

In Exercises 105–110, use an associative property to write an algebraic expression equivalent to each expression and simplify. 

-1/3(-3y)

In Exercises 33–68, add or subtract as indicated. 3/(2x+4) + 2/(3x+6)

Rational Expressions and Domain

In Exercises 177–178, insert parentheses to make each statement true.

8-2•3-4=14

In Exercises 42–46, simplify each algebraic expression.

3(4y-5) - (7y+2)

Find the domain of each rational expression. 3 / x2 - 5x - 6

Add or subtract, as indicated. 1/6m + 2/5m + 4/m

Simplify each complex fraction. [ 6/(x^2-25) + x ] / [ 1/(x - 5) ]

Simplify each complex fraction. [ 1/(a^3+b^3) ] / [ 1/(a^2 + 2ab + b^2) ]

In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s).

7+5x, for x=10

Add or subtract as indicated. [(2x-7)/(x^2-9)]-[(x-10)/(x^2-9)]

Perform the indicated operation, and write each answer in lowest terms. y^3/8 ÷ y/4

Write each rational expression in lowest terms. 8k + 16 / 9k + 18

In Exercises 83–100, use the order of operations to simplify each expression.

3(-2)² - 4(-3)²

In Exercises 27–50, simplify each rational expression. If the rational expression cannot be simplified, so state.

(x+2)/(x²−x−6)

In Exercises 85–96, simplify each algebraic expression. 5(3x+4)−4

Simplify each complex fraction. [ 1/(y+3) - 1/y ] / (1/y)

Rewrite simple rational expressions using inspection

Perform the indicated operation, and write each answer in lowest terms. 4/x-y - 9/x-y

In Exercises 22–33, perform the indicated operation.

-3/5 - (-1/2)

In Exercises 27–50, simplify each rational expression. If the rational expression cannot be simplified, so state.

(x²+7x−18)/(x²−3x+2)

In Exercises 105–110, use an associative property to write an algebraic expression equivalent to each expression and simplify.

-7(3x)

In Exercises 35–46, subtract as indicated.

1/4-1/2

In Exercises 73–90, divide as indicated.

4/y−6 ÷ 40/7y−42

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

Five more than a number

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

Four more than five times a number

In Exercises 83–100, use the order of operations to simplify each expression.

20+1-√(10²-(5+1)²) (-2)

Simplify each complex fraction. [ 1/(x+1) - 1/x ] / (1/x)

Evaluate each expression for x = -4 and y = 2. 2|y| - 3|x| / |xy|

In Exercises 111–116, use the distributive property to write an equivalent expression..

-7(2x+3)

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

Nine less than a number

Add or subtract as indicated. 12/25x^2+11/15x^3

In Exercises 27–50, simplify each rational expression. If the rational expression cannot be simplified, so state.

(x²+3xy−10y²)/(3x²−7xy+2y²)

Evaluate each expression for x = -4 and y = 2. |-8y + x| / -|x|

In Exercises 51–72, multiply as indicated.

2x²−3xy−2y²/3x²−4xy+y² ⋅ 3x²−2xy−y²/x²+xy−6y²

Write each rational expression in lowest terms. 8m^2 + 6m - 9 / 16m^2 - 9

In Exercises 71–82, divide as indicated or state that the division is undefined.

12 / -4

Multiply or divide, as indicated. (xz - xw + 2yz - 2yw)/(z^2 - w^2) * (4z + 4w + xz + wx)/(16 - x^2)

In Exercises 35–41, simplify each expression.

-11 - [-17+(-3)]

In Exercises 33–68, add or subtract as indicated. x/(x^2−2x−24) − x/(x^2−7x+6)

❖ Finding the Domain of a Function Algebraically (No graph!) ❖

Evaluate each expression for x = -4 and y = 2. |-5y + x|

Evaluate each expression for x = -4 and y = 2. |3x - 2y|

In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s).

6x-y, for x=3 and y=8

In Exercises 1–3, write each English phrase as an algebraic expression. Let x represent the number.

The quotient of nine and a number, increased by half of the number

Perform all indicated operations, and write each answer with positive integer exponents. [ (x^-2 + y^-2)/ (x^-2 - y^-2) ] * [ (x+y)/(x-y) ]

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). 7+5x, for x = 10

In Exercises 19–21, find each absolute value.

|5.003|

In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 

The quotient of three and the difference of five and a number

Find the domain of each rational expression. x^3 - 1 / x - 1

In Exercises 27–50, simplify each rational expression. If the rational expression cannot be simplified, so state.

(x²−4)/(x−2)

Finding the Domain of an Expression Involving Fractions - Example 2

Use specific values for x and y to show that, in general, 1/x + 1/y is not equivalent to 1 / x + y.

In Exercises 101–104, write an algebraic expression equivalent to the given expression using each of the commutative properties.

4x+10

In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s).

8x-y, for x=3 and y=4

Perform all indicated operations, and write each answer with positive integer exponents. {x- 9y^-1}/{ (x-3y^-1)(x+3y^-1)}

Simplifying Complex Fractions - Ex 2

In Exercises 51–72, multiply as indicated.

x−3/x+7 ⋅ 3x+21/2x−6

In Exercises 85–96, simplify each algebraic expression. 2(5x−1)+14x

Manipulating Rational Expressions: Simplification and Operations

In Exercises 73–90, divide as indicated.

(x²−16) ÷ x²+3x−4/x²+4

In Exercises 51–72, multiply as indicated.

8y+2/y²−9 ⋅ 3−y/4y²+y

Add or subtract, as indicated. 3/2k + 5/3k

Add or subtract, as indicated. (17y + 3)/(9y + 7) - (-10y - 18)/(9y + 7)

Exercises 97–99 will help you prepare for the material covered in the first section of the next chapter.

Evaluate 5x+7 for x = a + h.

In Exercises 69–82, simplify each complex rational expression. (x/3−1)/(x−3)

In Exercises 29–34, find -x for the given value of x.

x=11

Simplify each complex fraction. [ (x+4)/x - 3/(x-2) ] / [ x/(x - 2) + 1/x ]

In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s).

x²+2x, for x=1/2

In Exercises 111–120, use the order of operations to simplify each expression. 8−3[−2(2−5)−4(8−6)]

Multiply rational expressions

Write each rational expression in lowest terms. m^2 - 4m + 4 / m^2 + m - 6

Simplify each complex fraction. [ m - 1/(m^2-4) ] / [ 1/(m + 2) ]

Multiply or divide as indicated. [(x^2-5x-24)/(x^2-x-12)]/[(x^2-10x+16)/(x^2+x-6)]

In Exercises 47–58, multiply as indicated.

(-4)(-2)(-1)

In Exercises 42–46, simplify each algebraic expression.

6(2x-3) - 5(3x-2)

Rational Expressions: Writing in Lowest Terms - Ex 1

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). (2x+3y)/(x+1), for x=-2 and y=4

Simplify each complex fraction. [ y + 1/(y^2-9) ] / [ 1/(y + 3) ]

Simplify each complex fraction. [ (-2)/(x+h) - (-2)/x ] / h

Find the domain of each rational expression. 
x^2 - 1 / x + 1