Simplifying
Multiplying and dividing
Adding and subtracting
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1
Mr. Salin
Integrated Algebra
What is a rational expression:
Example:
𝑥 2 −9
𝑥 2 +7𝑥+10
What would make a rational expression undefined?
What value of x would make the denominator equal to zero?
Solve this algebraically by setting the denominator to zero
and solve.
𝑥 2 −9
𝑥 2 +7𝑥+10
x2 + 7x + 10 = 0
(
)(
)=0
Either
The values of x that would make the rational
expression undefined are {
}
What values of x would make the expression
𝑥 2 −3𝑥−10
𝑥 2 +6𝑥+8
undefined?
What values of x would make the expression
𝑥 2 −7𝑥−12
undefined?
𝑥 2 −36
What values of x would make the expression
𝑥 2 +3𝑥+2
16−𝑥 2
undefined?
undefined?
𝑥 2 −3𝑥−10
Page
𝑥 2 −5𝑥−4
2
What values of x would make the expression
Find the undefined values of the variable for each
rational expression.
1.
3.
7
x 3
x 9
x  x  12
2
2.
x 3
x2  9
4.
x 3
x  5x  6
2
Simplifying a rational expression:
Goal: Factor and reduce to lowest terms
Factor the Numerator & denominator by:
1st – look to “pull out” the GCF
2nd – Factor by reverse distributive property
Simplify:
Simplify:
2
=
3𝑥 2 +6𝑥
𝑥+2
=
𝑥 2 +6𝑥+8
2𝑥+8
=
𝑥 2 −4𝑥−21
𝑥 2 −9
=
3
Simplify:
4𝑥+6
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Simplify:
Simplify:
5𝑥 2 +5𝑥−10
15𝑥 2 +60𝑥−75
=
 18 x 2
12 x 4
2.
x ( x 2  6)
x2
3.
2x 2  6 x
6x 2
4.
42x  6 x 3
36 x
5.
4  x2
x2  x  2
6.
x 2  25
2 x  10
7.
x 2  x  20
x 2  2 x  15
8.
x3  x
x 3  5x 2  6x
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PRACTICE: Simplify the expression if possible.
Multiplying and dividing rational expressions:
Step 1: Follow the rules for multiplying and dividing fractions
To make one fraction.
Step 2: Simplify by factoring the GCF
Step 3: Simplify by reverse FOIL
Example: Multiply
9 2
∙ =
11 3
Find the Product
3𝑥 7𝑥 2
∙
=
14𝑥 2 15𝑥
Find the Product
𝑥2 − 4
9
∙
=
3
𝑥+2
Find the Product
4𝑥 + 12 𝑥 2 + 6𝑥
∙
=
𝑥+6
𝑥+3
Find the quotient
5 15
÷
=
9 36
Find the quotient
2𝑥
𝑥
÷
=
𝑥+3 𝑥+3
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Find the quotient
𝑥 2 − 2𝑥 − 8 2𝑥 + 4
÷ 2
=
𝑥 2 − 4𝑥
𝑥 −𝑥
PRACTICE PROBLEMS:
1.
9x 2 8

4 18 x
2.
3x 3 8x 2

4 x 15 x 4
3.
16 x 2 4 x 2

8x
16 x
4.
25 x 2 5 x

10 x 10 x
5.
4x
x 3
 2
x  9 8 x  12 x
6.
3x
x 6
 2
x  2 x  24 6 x  9 x
7.
5 x  15 x  3

3x
9x
8.
x 2  4x  3 x  1

2x
2
9.
4 x 2  25
 ( 2 x  5)
4x
10.
2
6
x2  x  2
x 2  2x  3

x 2  5 x  6 x 2  7 x  12
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Adding & Subtracting Rational Expressions:
Step 1: Must have common denominators
Remember to change the numerator if you change the
denominator
Step 2: Make one fraction by combining numerators,
denominator stays the same
Step 3: Combine like terms in the numerator
Step 4: Simplify
Add:
5
7
+
=
3𝑥 3𝑥
Subtract:
3𝑥
𝑥+5
−
=
𝑥−1 𝑥−1
Add:
3
4
+
=
2𝑥 3𝑥
Subtract
9
8
− 2=
3𝑥 4𝑥
Find the Sum:
3
2𝑥
+
=
𝑥+1 𝑥−3
7 x2

2x
2x
2.
7x 6x

x3 x3
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PRACTICE PROBLEMS: Simplify the expression.
3.
4
2x  2

x 1 x 1
4.
2
5x

3x  1 3x  1
5.
5X
20

x4 4x
6.
x
3x  1
 2
x 9 x 9
7.
11
2

6 x 12 x
8.
9
2
 2
5x x
9
3

x  3x x  3
2
LCD:
10.
4
7

x 4 x 2
LCD:
8
9.
LCD:
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LCD:
2
Solving Rational Equations (Proportions)
- Has an equal sign
- Need to solve for the variable
Step1: Cross Multiply
Step 2: Gather x’s on one side
Step 3: If you have an x2 , gather everything on one side, set
the other to zero and factor.
Solve for x:
𝑥+4 𝑥+9
=
2
3
Solve for x:
5 𝑥 + 13
=
𝑥
6
Solve for x:
6
𝑥
=
𝑥+4 2
Solve for x:
2
26
−3=
𝑥
𝑥
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9
Solve for x:
𝑥 − 3 2𝑥 + 3
=
𝑥+6
𝑥+6
Practice: Solve for x:
5.
7.
9.
𝑥+1
3
𝑥−13
2𝑥
4−𝑥
𝑥−11
𝑥
=
=
𝑥
𝑥
𝑥
20
𝑥
3
𝑥−11
8.
=
2
2𝑥
6.
𝑥−4
=
𝑥−3
4.
10
+1=
𝑥−1
2.
𝑥−3
=
6𝑥
5
2
=
𝑥−3
𝑥
𝑥+7
10.
2
𝑥−3
15
𝑥+1
=
24
𝑥
−3=
2𝑥
5
−1
𝑥+7
1
7𝑥−2
3
15
+ =
10
3.
3
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