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Unit 3: Making Sense of Rational Expressions

Open Posted By: surajrudrajnv33 Date: 21/01/2021 Graduate Assignment Writing

 

Making Sense of Rational Expressions

A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. Here are some examples of rational expressions. The last one may look a little strange since it is more commonly written. Simplifying rational expressions requires good factoring skills. The twist now is that you are looking for factors that are common to both the numerator and the denominator of the rational expression.

This unit emphasizes performing mathematical operations on rational expressions and using these operations to solve equations and inequalities.

Number Sense, Concepts, and Operations

Add, subtract, multiply, and divide real numbers, including square roots and exponents, using appropriate methods of computing, such as mental mathematics, paper and pencil, and calculator. Describe, analyze and generalize relationships, patterns, and functions using words, symbols, variables, tables, and graphs. Determine the impact when changing the parameters of given functions. 

Category: Mathematics & Physics Subjects: Calculus Deadline: 12 Hours Budget: $120 - $180 Pages: 2-3 Pages (Short Assignment)

Attachment 1

Unit 3: Making Sense of Rational Expressions

Unit Focus

Algebraic Thinking

This unit emphasizes performing mathematical operations on rational expressions and using these operations to solve equations and inequalities.

Number Sense, Concepts, and Operations

• Add, subtract, multiply, and divide real numbers, including square roots and exponents, using appropriate methods of computing, such as mental mathematics, paper and pencil, and calculator. (MA.A.3.4.3)

• Describe, analyze and generalize relationships, patterns and functions using words, symbols, variables, tables, and graphs. (MA.D.1.4.1)

• Determine the impact when changing parameters of given functions. (MA.D.1.4.2)

Unit 3: Making Sense of Rational Expressions 165

Unit 3: Making Sense of Rational Numbers

Introduction

Algebra students must be able to add, subtract, multiply, divide, and simplify rational expressions efficiently. These skills become more important as you become more involved in using mathematics. As a liberal arts mathematics student, you will have the opportunity to reacquaint yourself with the topics and methods you will need for future mathematical success.

Lesson One Purpose

• Add, subtract, multiply, and divide real numbers, including square roots and exponents, using appropriate methods of computing, such as mental mathematics, paper and pencil, and calculator. (MA.A.3.4.3)

• Describe, analyze and generalize relationships, patterns and functions using words, symbols, variables, tables, and graphs. (MA.D.1.4.1)

Unit 3: Making Sense of Rational Expressions166

Simplifying Rational Expressions

An expression is a collection of numbers, symbols, and/or operation signs that stand for a number. A fraction, or any part of a whole, is an expression that represents a quotient—the result of dividing two numbers. The same fraction may be expressed in many different ways.

1 2 =

2 4 =

3 6 =

5 10

If the numerator (top number) and the denominator (bottom number) are both polynomials, then we call the fraction a rational expression. A rational expression is a fraction whose numerator and/or denominator are polynomials. The fractions below are all rational expressions.

When the variables or any symbols which could represent numbers (usually letters) are replaced, the result is a numerator and a denominator that are real numbers. In this case, we say the entire expression is a real number. Real numbers are all rational numbers and irrational numbers— real numbers that cannot be expressed as a ratio of two integers. Of course, there is an exception…

when a denominator is equal to 0, we say the fraction is undefined.

Note: In this unit, we will agree that no denominator equals 0.

x + y a2 – 2a + 1x

a y2 + 4 1

b – 3 a

Unit 3: Making Sense of Rational Expressions 167

Fractions have some interesting properties. Let’s examine them.

• If ab = c d , then ad = bc.

4 8 =

6 12 therefore 4 • 12 = 8 • 6

• ab = ac bc

4 7 =

4 • 3 7 • 3 therefore

4 7 =

12 21

Simply stated, if you multiply both the numerator and the denominator by the same number, the new fraction will be equivalent to the original fraction.

• acbc = a b

9 21 =

9 ÷ 3 21 ÷ 3 therefore

9 21 =

3 7

Stated in words, if you divide both the numerator and the denominator by the same number, the new fraction will be equivalent to the original fraction. The same rules are true for simplifying rational expressions by performing as many indicated operations as possible. Many times, however, it is necessary to factor and find numbers or expressions that divide the numerator or the denominator, or both, so that the common factors become easier to see. Look at the following example:

3 3x + 3y = 3

3(x + y)

1

1 = x + y

Notice that by factoring a 3 out of the numerator, we can divide (or cancel) the 3s, leaving x + y as the final result.

Before we move on, do the practice on the following pages.

a b

c d=

=a • d b • c =ad bc

In other words, if two fractions are equal, then the products are equal when you cross multiply.

4 8

6 12=

=4 • 12 8 • 6 =48 48

Unit 3: Making Sense of Rational Expressions172

Additional Factoring

Look carefully at numbers 2-5 in the previous practice. What do you notice about them?

Alert! You cannot cancel individual terms (numbers, variables, products, or quotients in an expression)—you can only cancel factors (numbers or expressions that exactly divide another number)!

2x + 4 ≠ 4

2x 4

3x + 6 ≠ 3

x + 6 3

9x2 + 3 ≠ 9x 2x

6x6x + 3

Look at how simplifying these expressions was taken a step further. Notice that additional factoring was necessary.

Example

Look at the denominator above. It is one of the factors of the numerator. Often, you can use the problem for hints as you begin to factor.

= x + 3

(x + 3)(x + 2) = (x + 2) = x + 2 1x2 + 5x + 6

x + 31

Unit 3: Making Sense of Rational Expressions180

Lesson Two Purpose

• Add, subtract, multiply, and divide real numbers, including square roots and exponents, using appropriate methods of computing, such as mental mathematics, paper and pencil, and calculator. (MA.A.3.4.3)

• Describe, analyze and generalize relationships, patterns and functions using words, symbols, variables, tables, and graphs. (MA.D.1.4.1)

Addition and Subtraction of Rational Expressions

In order to add and subtract rational expressions in fraction form, it is necessary for the fractions to have a common denominator (the same bottom number). We find those common denominators in the same way we did with simple fractions. The process requires careful attention.

• When we add 37 + 5 8 , we find a common denominator

by multiplying 7 and 8.

• Then we change each fraction to an equivalent fraction whose denominator is 56.

3 • 8 = 2456 and7 • 8 5 • 7 = 35568 • 7

• Next we add 2456 + 35 56 =

59 56 .

Unit 3: Making Sense of Rational Expressions 181

Finding the Least Common Multiple (LCM)

By multiplying the denominators of the terms we intend to add or subtract, we can always find a common denominator. However, it is often to our advantage to find the least common denominator (LCD), which is also the least common multiple (LCM). The LCD or LCM is the smallest of the common multiples of two or more numbers. This makes simplifying the result easier. Look at the example below.

Let’s look at finding the LCM of 36, 27, and 15.

1. Factor each of the denominators and examine the results.

36 = 2 • 2 • 3 • 3 The new denominator must contain at least two 2s and two 3s.

27 = 3 • 3 • 3 The new denominator must contain at least three 3s.

45 = 3 • 3 • 5 The new denominator must contain at least two 3s and one 5.

2. Find the minimum combination of factors that is described by the combination of all the statements above—two 2s, three 3s, and one 5.

LCM = 2 • 2 • 3 • 3 • 3 • 5 = 540

two 2s three 3s one 5

3. Convert the terms to equivalent fractions using the new common denominator and then proceed to add or subtract.

5 36 =

75 540 ;

8 27 =

160 540 ;

4 15 =

144 540

75 540 +

160 540 –

144 540 =

91 540

Unit 3: Making Sense of Rational Expressions182

Now, let’s look at an algebraic example.

1 =

y2 – 4y – 21 –

y y2 – 9

1. Factor each denominator and examine the results.

y2 – 9 = (y + 3)(y – 3) The new denominator must contain (y + 3) and (y – 3).

y2 – 4y – 21 = (y – 7)(y + 3) The new denominator must contain (y – 7) and (y + 3).

2. Find the minimum combination of factors.

LCM = (y + 3)(y – 3)(y – 7)

3. Convert each fraction to an equivalent fraction using the new common denominator and proceed to subtract.

Hint: Always check to see if the numerator can be factored and then reduce, if possible.

notice how the minus sign

between the fractions

distributes to make

-y + 3 in the numerator

(distributive property) =

y2 – 8y + 3

y2 – 7y – y + 3

=– 1(y – 3)

(y + 3)(y – 3)(y – 7)

y(y – 7) (y + 3)(y – 3)(y – 7)

(y + 3)(y – 3)(y – 7)

(y + 3)(y – 3)(y – 7)

Unit 3: Making Sense of Rational Expressions 189

Lesson Three Purpose

• Add, subtract, multiply, and divide real numbers, including square roots and exponents, using appropriate methods of computing, such as mental mathematics, paper and pencil, and calculator. (MA.A.3.4.3)

• Describe, analyze and generalize relationships, patterns and functions using words, symbols, variables, tables, and graphs. (MA.D.1.4.1)

Multiplication and Division of Rational Expressions

To multiply fractions, you learned to multiply the numerators together, then multiply the denominators together, and then reduce, if possible.

3 8 x

5 7 =

15 56

We use this same process with rational expressions.

4 5x 13

11x =x = 65 44

65x 44x

Sometimes it is simpler to reduce or cancel common factors before multiplying.

13 11x4

5x 13 11x 4

5x=x x = 65 44

When we need to divide fractions, we invert (flip over) the factor to the right of the division symbol and then multiply.

3y 5y3 4x

=÷ • = 2x2

3y 5y3

4x 2x2 10x2y3

12xy =

5xy2

6

invert

Unit 3: Making Sense of Rational Expressions190

Pay careful attention to negative signs in the factors.

Decide before you multiply whether the answer will be positive or negative.

• If the number of negative factors is even, the result will be positive.

• If the number of negative factors is odd, the answer will be negative.

Remember: In this unit, we agreed that no denominator equals 0.

Unit 3: Making Sense of Rational Expressions198

Lesson Four Purpose

• Add, subtract, multiply, and divide real numbers, including square roots and exponents, using appropriate methods of computing, such as mental mathematics, paper and pencil, and calculator. (MA.A.3.4.3)

• Describe, analyze and generalize relationships, patterns and functions using words, symbols, variables, tables, and graphs. (MA.D.1.4.1)

• Determine the impact when changing parameters of given functions. (MA.D.1.4.2)

Solving Equations

An equation is a mathematical sentence that uses an equal sign to show that two quantities are equal. An equation equates one expression to another.

3x – 7 = 8 is an example of a equation.

You may be able to solve this problem mentally, without using paper and pencil.

3x – 7 = 8

The problem reads—3 times what number minus 7 equals 8?

Think: 3 • 4 = 12 12 – 7 = 5 too small

Think: 3 • 5 = 15 15 – 7 = 8 That’s it!

3x – 7 = 8 3(5) – 7 = 8

Unit 3: Making Sense of Rational Expressions200

Step-by-Step Process for Solving Equations

A problem like 5 x + 12 = -2(x – 10) is a bit more challenging. You could use a

guess and check process, but that would take more time, especially when answers involve decimals or fractions.

So, as problems become more difficult, you can see that it is important to have a process in mind and to write down the steps as you go.

Unfortunately, there is no exact process for solving equations. Every rule has an exception. That is why practice is necessary and keeping a written record of the steps you have used is extremely helpful.

Example 1

Let’s look at a step-by-step process for solving the problem above.

5 x + 12 = -2(x – 10) Step 1: Copy the problem carefully!

Step 2: Simplify each side of the equation as needed by distributing the 2.

Step 3: Multiply both sides of the equation by 5 to “undo” the division by 5, which eliminates the fraction.

Step 4: Simplify by distributing the 5.

Step 5: Add 10x to both sides.

Step 6: Subtract 12 from both sides.

Step 7: Divide both sides by 11.

Step 8: Check by replacing the variable in the original problem.

It checks!

5 x + 12 = -2x + 20

5 x + 12 • 5 = (-2x + 20) • 5( )

x + 12 = -10x + 100

+ 10x + 1x + 12 = -10x + 10x + 100 11x + 12 = 100

11x + 12 – 12 = 100 – 12 11x = 88

11x ÷ 11 = 88 ÷ 11 x = 8

5 x + 12 = -2(x – 10)

5 8 + 12 = -2(8) + 20

4 = -16 + 20 4 = 4

Unit 3: Making Sense of Rational Expressions 201

Example 2

What if the original problem had been 5x + 12 = -2(x – 10)? The process would have been different. Watch for differences.

5x + 12 = -2(x – 10) Step 1: Copy the problem carefully!

Step 2: Simplify each side of the equation as needed by distributing the 2.

Step 3: Subtract 12 from both sides of the equation.

Step 4: Add 2x to both sides of the equation.

Step 5: Divide both sides by 7.

Step 6: Check by replacing the variable in the original problem.

It checks!

5x + 12 = -2x + 20

5x = -2x + 8

5x + 2x = -2x + 2x + 8 7x = 8

7x ÷ 7 = 8 ÷ 7 x =

17 = 17

5x + 12 – 12 = -2x + 20 – 12

7 8

5x + 12 = -2(x – 10)

5( ) + 12 = -2( ) + 207 8

7 8

7 40 + 12 = + 20

7 -16

7 5 5 + 12 = -2 + 207

2

7 5

7 5

Did you notice that the steps were not always the same? The rules for solving equations change to fit the individual needs of each problem. You can see why it is a good idea to check your answers each time. You may need to do some steps in a different order than you originally thought.

Unit 3: Making Sense of Rational Expressions202

Generally speaking the processes for solving equations are as follows.

• Simplify both sides of the equation as needed.

• “Undo” additions and subtractions.

• “Undo” multiplications and divisions.

You might notice that this seems to be the opposite of the order of operations. Typically, we “undo” in the reverse order from the original process.

Guidelines for Solving Equations

1. Use the distributive property to clear parentheses.

2. Combine like terms. We want to isolate the variable.

3. Undo addition or subtraction using inverse operations.

4. Undo multiplication or division using inverse operations.

5. Check by substituting the solution in the original equation.

SAM = Simplify (steps 1 and 2) then Add (or subtract) Multiply (or divide)

Unit 3: Making Sense of Rational Expressions 203

Here are some additional examples.

Example 3

Solve:

6y + 4(y + 2) = 88 6y + 4y + 8 = 88 use distributive property 10y + 8 – 8 = 88 – 8 combine like terms and undo addition

by subtracting 8 from each side 10y 10

= 8010 undo multiplication by dividing y = 8 by 10

Check solution in the original equation:

6y + 4(y + 2) = 88 6(8) + 4(8 + 2) = 88

48 + 4(10) = 88 48 + 40 = 88

88 = 88 It checks!

Example 4

Solve:

- 12 (x + 8) = 10

- 12 x – 4 = 10 use distributive property

- 12 x – 4 + 4 = 10 + 4 undo subtraction by adding 4 to both sides

- 12 x = 14 (-2)- 12 x = 14(-2) isolate the variable by multiplying

x = -28 each side by the reciprocal of - 12

Check solution in the original equation:

- 12 (x + 8) = 10

- 12 (-28 + 8) = 10

- 12 (-20) = 10 10 = 10 It checks!

Unit 3: Making Sense of Rational Expressions204

Example 5

Solve:

26 = 23 (9x – 6)

26 = 23 (9x) – 2 3 (6) use distributive property

26 = 6x – 4 26 + 4 = 6x – 4 + 4 undo subtraction by adding 4 to

each side 30 6 =

6x 6 undo multiplication by dividing

each side by 6 5 = x

Check solution in the original equation:

26 = 23 (9x – 6)

26 = 23 (9 • 5 – 6)

26 = 23 (39) 26 = 26 It checks!

Unit 3: Making Sense of Rational Expressions 205

Example 6

Solve:

x – (2x + 3) = 4 x – 1(2x + 3) = 4 use the multiplicative property of -1

x – 2x – 3 = 4 use the multiplicative identity of 1 and use the distributive property

-1x – 3 = 4 combine like terms -1x – 3 + 3 = 4 + 3 undo subtraction

-1x -1 =

7 -1 undo multiplication

x = -7

Examine the solution steps above. See the use of the multiplicative property of -1 in front of the parentheses on line two.

line 1: x – (2x + 3) = 4 line 2: x – 1(2x + 3) = 4

Also notice the use of multiplicative identity on line three.

line 3: 1x – 2x – 3 = 4

The simple variable x was multiplied by 1 (1 • x) to equal 1x. The 1x helped to clarify the number of variables when combining like terms on line four.

Check solution in the original equation:

x – (2x + 3) = 4 -7 – (2 • -7 + 3) = 4

-7 – (-11) = 4 4 = 4 It checks!

Unit 3: Making Sense of Rational Expressions214

Lesson Five Purpose

• Add, subtract, multiply, and divide real numbers, including square roots and exponents, using appropriate methods of computing, such as mental mathematics, paper and pencil, and calculator. (MA.A.3.4.3)

• Describe, analyze and generalize relationships, patterns and functions using words, symbols, variables, tables, and graphs. (MA.D.1.4.1)

• Determine the impact when changing parameters of given functions. (MA.D.1.4.2)

Solving Inequalities

Inequalities are mathematical sentences that are not equal. Instead of using the equal symbol (=), we use the following with inequalities.

• greater than >

• less than <

• greater than or equal to ≥

• less than or equal to ≤

• not equal to ≠

Remember: The “is greater than” (>) or “is less than” (<) symbols always point to the lesser number.

For example:

5 3 3 5

> <

Unit 3: Making Sense of Rational Expressions 215

We have been solving equations in this unit. When we solve inequalities, the procedures are the same except for one important difference.

When we multiply or divide both sides of an inequality by the same negative number, we reverse the direction of the inequality symbol.

Example

Solve by dividing by a negative number and reversing the inequality sign.

-3x < 6 -3x -3 > -3

6 divide each side by -3 and reverse the inequality symbol

x > -2

To check this solution, pick any number greater than -2 and substitute your choice into the original inequality. For instance, -1, 0, or 3, or 3,000 could be substituted into the original problem.

Check with different solutions of numbers greater than -2:

substitute -1 substitute 3

-3x < 6 -3x < 6 -3(-1) < 6 -3(3) < 6

3 < 6 It checks! -9 < 6 It checks!

substitute 0 substitute 3,000

-3x < 6 -3x < 6 -3(0) < 6 -3(3,000) < 6

0 < 6 It checks! -9,000 < 6 It checks!

Notice that -1, 0, 3, and 3,000 are all greater than -2 and each one checks as a solution.

Unit 3: Making Sense of Rational Expressions216

Study the following examples.

Example

Solve by multiplying by a negative number and reversing the inequality sign.

- 13 y ≥ 4 (-3) - 13 y ≤ 4(-3) multiply each side by -3 and

reverse the inequality symbol y ≤ -12

Example

Solve by first adding, then dividing by a negative number, and reversing the inequality sign.

-3a – 4 > 2 -3a – 4 + 4 > 2 + 4 add 4 to each side

-3a > 6 -3a -3 < -3

6 divide each side by -3 and reverse the inequality symbol

a < -2

Example

Solve by first subtracting, then multiplying by a negative number, and reversing the inequality sign.

y -2 + 5 ≤ 0

y -2 + 5 – 5 ≤ 0 – 5 subtract 5 from each side

y -2 ≤ -5

(-2)y -2 ≥ (-5)(-2) multiply each side by -2 and

reverse the inequality symbol y ≥ 10

Unit 3: Making Sense of Rational Expressions 217

Example

Solve by first subtracting, then multiplying by a positive number. Do not reverse the inequality sign.

n 2 + 5 ≤ 2

n 2 + 5 – 5 ≤ 2 – 5 subtract 5 from each side

n 2 ≤ -3

(2)n 2 ≤ -3(2) multiply each side by 2, but n ≤ -6 do not reverse the inequality symbol because

we multiplied by a positive number

When multiplying or dividing both sides of an inequality by the same positive number, do not reverse the inequality symbol—leave it alone.

Example

Solve by first adding, then dividing by a positive number. Do not reverse the inequality sign.

7x – 3 > -24

7x – 3 + 3 > -24 + 3 add 3 to each side

7x > -21 7x 7 >

-21 7 divide each side by 7, but

x > -3 do not reverse the inequality symbol because we divided by a positive number

  • 3a LAM SB Unit 3 155-179.pdf
  • 3b LAM SB Unit 3 180-188.pdf
  • 3c LAM SB Unit 3 189-197.pdf
  • 3d LAM SB Unit 3 198-213.pdf
  • 3e LAM SB Unit 3 214-232.pdf

Attachment 2

Unit 3: Making Sense of Rational Expressions 157

Vocabulary

Use the vocabulary words and definitions below as a reference for this unit.

canceling ......................................... dividing a numerator and a denominator by a common factor to write a fraction in lowest terms or before multiplying fractions Example: 15 24 =

3 • 5 2 • 2 • 2 • 3 =

5 8

1

1

common denominator .................. a common multiple of two or more denominators Example: A common denominator for 14 and 56 is 12.

common factor ............................... a number that is a factor of two or more numbers Example: 2 is a common factor of 6 and 12.

common multiple .......................... a number that is a multiple of two or more numbers Example: 18 is a common multiple of 3, 6, and 9.

Unit 3: Making Sense of Rational Expressions158

cross multiplication ...................... a method for solving and checking proportions; a method for finding a missing numerator or denominator in equivalent fractions or ratios by making the cross products equal Example: To solve this proportion:

n 9

8 12

12 x n = 9 x 8 12n = 72

n = 7212 n = 6

Solution: 6 9

= 8 12

decimal number ............................ any number written with a decimal point in the number Example: A decimal number falls between two whole numbers, such as 1.5 falls between 1 and 2. Decimal numbers smaller than 1 are sometimes called decimal fractions, such as five-tenths is written 0.5.

denominator ................................... the bottom number of a fraction, indicating the number of equal parts a whole was divided into Example: In the fraction 23 the denominator is 3, meaning the whole was divided into 3 equal parts.

difference ........................................ a number that is the result of subtraction Example: In 16 – 9 = 7, 7 is the difference.

Unit 3: Making Sense of Rational Expressions 159

distributive property .................... the product of a number and the sum or difference of two numbers is equal to the sum or difference of the two products Example: x(a + b) = ax + bx

equation .......................................... a mathematical sentence in which two expressions are connected by an equality symbol Example: 2x = 10

equivalent

(forms of a number) ...................... the same number expressed in different forms Example: 34 , 0.75, and 75%

expression ....................................... a collection of numbers, symbols, and/or operation signs that stands for a number Example: 4r2; 3x + 2y; 25 Expressions do not contain equality (=) or inequality (<, >, ≤, ≥, or ≠) symbols.

factor ................................................ a number or expression that divides evenly into another number; one of the numbers multiplied to get a product Example: 1, 2, 4, 5, 10, and 20 are factors of 20 and (x + 1) is one of the factors of (x2 – 1).

factoring .......................................... expressing a polynomial expression as the product of monomials and polynomials Example: x2 – 5x + 4 = 0

(x – 4)(x – 1) = 0

Unit 3: Making Sense of Rational Expressions160

fraction ............................................ any part of a whole Example: One-half written in fractional form is 12 .

inequality ....................................... a sentence that states one expression is greater than (>), greater than or equal to (≥), less than (<), less than or equal to (≤), or not equal to (≠) another expression Example: a ≠ 5 or x < 7 or 2y + 3 ≥ 11

integers ........................................... the numbers in the set {… , -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

inverse operation .......................... an action that undoes a previously applied action Example: Subtraction is the inverse operation of addition.

irrational number .......................... a real number that cannot be expressed as a ratio of two integers Example: 2

least common denominator

(LCD) ............................................... the smallest common multiple of the denominators of two or more fractions Example: For 34 and

1 6 , 12 is the least

common denominator.

least common multiple

(LCM) .............................................. the smallest of the common multiples of two or more numbers Example: For 4 and 6, 12 is the least common multiple.

Unit 3: Making Sense of Rational Expressions 161

like terms ........................................ terms that have the same variables and the same corresponding exponents Example: In 5x2 + 3x2 + 6, 5x2 and 3x2 are like terms.

minimum ........................................ the smallest amount or number allowed or possible

multiplicative identity ................. the number one (1); the product of a number and the multiplicative identity is the number itself Example: 5 x 1 = 5

multiplicative property of -1 ...... the product of any number and -1 is the opposite or additive inverse of the number Example: -1(a) = -a and a(-1) = -a

negative numbers ......................... numbers less than zero

numerator ....................................... the top number of a fraction, indicating the number of equal parts being considered Example: In the fraction 23 , the numerator is 2.

order of operations ....................... the order of performing computations in parentheses first, then exponents or powers, followed by multiplication and/or division (as read from left to right), then addition and/or subtraction (as read from left to right); also called algebraic order of operations Example: 5 + (12 – 2) ÷ 2 – 3 x 2 =

5 + 10 ÷ 2 – 3 x 2 = 5 + 5 – 6 =

10 – 6 = 4

Unit 3: Making Sense of Rational Expressions162

polynomial ..................................... a monomial or sum of monomials; any rational expression with no variable in the denominator Examples: x3 + 4x2 – x + 8 5mp2

-7x2y2 + 2x2 + 3

positive numbers .......................... numbers greater than zero

product ............................................ the result of multiplying numbers together Example: In 6 x 8 = 48, 48 is the product.

quotient ........................................... the result of dividing two numbers Example: In 42 ÷ 7 = 6, 6 is the quotient.

ratio .................................................. the comparison of two quantities Example: The ratio of a and b is a:b or ab , where b ≠ 0.

rational expression ....................... a fraction whose numerator and/or denominator are polynomials

Examples: x2 + 1 4x2 + 1

x + 2 5x

8

rational number ............................ a real number that can be expressed as a ratio of two integers

real numbers .................................. the set of all rational and irrational numbers

reciprocals ...................................... two numbers whose product is 1; also called multiplicative inverses

Example: Since 43 = 3 4 x 1, the

reciprocal of 34 is 4 3 .

Unit 3: Making Sense of Rational Expressions 163

simplest form

(of a fraction) ................................. a fraction whose numerator and denominator have no common factor greater than 1 Example: The simplest form of 36 is

1 2 .

simplify an expression ................. to perform as many of the indicated operations as possible

solution ........................................... any value for a variable that makes an equation or inequality a true statement Example: In y = 8 + 9

y = 17 17 is the solution.

substitute ........................................ to replace a variable with a numeral Example: 8(a) + 3

8(5) + 3

sum .................................................. the result of adding numbers together Example: In 6 + 8 = 14, 14 is the sum.

term .................................................. a number, variable, product, or quotient in an expression Example: In the expression 4x2 + 3x + x, 4x2, 3x, and x are terms.

variable ........................................... any symbol, usually a letter, which could represent a number