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
