Unit 3: Making Sense of Rational Expressions 173

Practice

Factor each of these and then simplify. Look for hints within the problem. Refer

to the previous page as necessary. Show essential steps.

1.

2.

Sometimes, it is necessary to factor both the numerator and denominator.

Examine the example below, then simplify each of the following expressions.

Example:

Note: The x’s do

not cancel.

3.

4.

a2 – 3a + 2 = a – 2

b2 – 2b – 3 = b – 3

x2 – 4 = x2 + x – 6

(x + 2)(x – 2)

(x + 3)(x – 2) = (x + 2)

(x + 3)

1

1

= x + 2

x + 3

2r

2 + r – 6 = r

2 + r – 2

x2 + x – 2 = x2 – 1

174 Unit 3: Making Sense of Rational Expressions

Practice

Simplify each expression. Show essential steps.

1.

2.

3.

4.

5b – 10 = b – 2

6a – 9 = 10a – 15

9x + 3 = 9

6b + 9 = 12

Unit 3: Making Sense of Rational Expressions 175

5.

6.

7.

8.

3a2

b + 6ab – 9b2

= 3b

x2 – 16 = x + 4

2a – b = b2 – 4a2

6×2 + 2 = 9×2 + 3

176 Unit 3: Making Sense of Rational Expressions

Practice

Factor each of these expressions and then simplify. Show essential steps.

1.

2.

3.

4.

y2

+ 5y – 14 = y + 2

a2

– 5a + 4 = a – 4

6m2

– m – 1 = 2m2

+ 9m – 5

2×2

+ x – 6 = 4×2 – 9

Unit 3: Making Sense of Rational Expressions 177

Practice

Use the list below to write the correct term for each definition on the line provided.

denominator

expression

fraction

rational expression

real numbers

variable

________________________ 1. a collection of numbers, symbols,

and/or operation signs that stands for a

number

________________________ 2. the top number of a fraction, indicating

the number of equal parts being

considered

________________________ 3. the bottom number of a fraction,

indicating the number of equal parts a

whole was divided into

________________________ 4. the set of all rational and irrational

numbers

________________________ 5. any part of a whole

________________________ 6. a fraction whose numerator and/or

denominator are polynomials

________________________ 7. any symbol, usually a letter, which

could represent a number

________________________ 8. a monomial or sum of monomials; any

rational expression with no variable in

the denominator

________________________ 9. the result of dividing two numbers

numerator

polynomial

quotient

178 Unit 3: Making Sense of Rational Expressions

Practice

Use the list below to complete the following statements.

canceling

cross multiplication

equivalent

factor

integers

product

simplify an expression

terms

1. If you multiply both the numerator and the denominator by the same

number, the new fraction will be because

it is the same number expressed in a different form.

2. are the numbers in the set

{… , -4, -3, -2, -1, 0, 1, 2, 3, 4, …}.

3. If you divide a numerator and a denominator by a common factor to

write a fraction in lowest terms or before multiplying fractions, you

are .

4. To , you need to perform as many of the

indicated operations as possible.

5. Numbers, variables, products, or quotients in an expression are

called .

Unit 3: Making Sense of Rational Expressions 179

6. A is a number or expression that divides

evenly into another number.

7. When you multiply numbers together, the result is called the

.

8. To find a missing numerator or denominator in equivalent fractions

or ratios, you can use a method called

and make the cross products equal.

206 Unit 3: Making Sense of Rational Expressions

Practice

Solve and check each equation. Use the examples on pages 200-205 for

reference. Show essential steps.

Hint: Find a step that looks similar to the problem you need help with and

follow from that point.

Remember: To check your work, replace the variable in the

original problem with the answer you found.

1. 3x – 7 = 17

2. 4x + 20 = x – 4

3. x

6 = 1.5

4. 2x

5 = 3.2

Unit 3: Making Sense of Rational Expressions 207

5. 5(x – 4) = 20

6. 5(4x – 7) = 0

7. 8x – 2x = 42

8. 5x – 3 = 2x + 18

9. -2x + 4 = -4x – 10

208 Unit 3: Making Sense of Rational Expressions

Practice

Solve and check each equation. Use the examples on pages 200-205 for

reference. Show essential steps.

1. 2(3x – 4) + 6 = 10

2. 3(x – 7) – x = -9

3. 2

3 x = 1

Hint: 2

3 x =

2x

3 . Rewrite 1 as 1

1 and cross multiply.

4. -1

2 x – 3

4 = 4

Unit 3: Making Sense of Rational Expressions 209

5. -3x = -33

8

6. -2

x = 8

7. -3x – 3

2 = 11

2

210 Unit 3: Making Sense of Rational Expressions

Practice

Solve and check each equation.

1. -87 = 9 – 8x

2. 4k + 3 = 3k + 1

3. 5a + 9 = 64

4. b

3 + 5 = -2

Unit 3: Making Sense of Rational Expressions 211

5. 4x = -(9 – x)

6. x

5 = -10

7. 3x – 1 = -x + 19

212 Unit 3: Making Sense of Rational Expressions

Practice

Solve and check each equation. Reduce fractions to simplest form.

1. 5x – 3 = 2x + 18

2. 6x – (4x – 12) = 3x + 5

3. x

6 = -24

5

4. 4(x – 2) = -3(x + 5)

Unit 3: Making Sense of Rational Expressions 213

5. 5( 1

3 x – 2) = 4

6. x

4 + 3

2 = 5

8

7. 9

2

x = 5

1

8. -1

2 + 8x

5 = -7

8

## HIS 255 Research Paper Outline with Updated Annotated Bibliography essay

student I’m studying and need help with a History question to help me learn. After continued research, construct a working outline for the research paper. The outline should cover the topics and supporting details of your research paper. The research paper must affirm the thesis on the approved Civil War topic you have selected. The […]