Pre-AP Algebra 2
Unit 8 - Lesson 8 – Solving rational equations
Objectives:
 Students will be able to solve rational equations and identify extraneous solutions.
Materials: Homework #8-7 answers overhead; tally sheets; pair work sheets; note-taking templates; homework #8-8
Time
5 min
10 min
15 min
25 min
Activity
Check Homework
Students check answers on the overhead. Pass around tally sheets.
Review Homework
Review the top 2-3 problems from the tally sheet.
Problems to grade: 1, 2a, 2b, 2c, 2d
Pair Work
Students graph functions on their calculators to estimate solutions to rational equations.
Discuss as needed.
Direct Instruction
Background
3
3
x  2  x  1 . This is most easily done by multiplying by the LCD.
4
5
Solve
Concepts
To solve a rational equation:
1) Multiply both sides by the LCD.
2) Solve.
3) Check your solutions in the original equation.
Examples
1) 1 5  20  4 . When you multiply by x2, you get x2 – 5x – 20 = 4. Combine like terms and factor.
x
x2
x2
Check the solutions (-3 and 8) in the original equation.
2)  2  x  8 . Make sure to highlight the step:
x 1
x 1
3) 5x  7  10
x2
25 min
x2
2(x  1)  (x  8)(x  1)
. This has x = 2 as an extraneous solution, so there is no solution.
Concepts
If the solution creates a division by 0 in the original equation, it is an extraneous (false) solution, and
should be crossed out. What does this mean graphically? Look at the graph and see that there is no
intersection.
Cross multiplication
Given a  c , multiply by bd. This yields abd  cbd   ad  cb .
b
d
b d
Whenever you have 2 fractions equal to each other, cross multiplication will give you an equivalent
equation. Note how this worked in example 2.
Pair Work
Students practice solving rational equations.
Homework #8-8: Quiz 2 Review
Pre-AP Algebra 2
8-8 Pair Work
Name: __________________________
Using the Calculator to Estimate
Solutions of Rational Equations
In order to solve the equation 2x + 5 = 5x – 7 graphically, you can graph f(x) = 2x + 5 and g(x) = 5x – 7,
and see where they intersect. Look at the graph that follows. Based on what you see, estimate the solution
for x.
x ≈ ______
Now, use algebra to solve the equation. Do you get the same
solution?
This method works for all functions, not just linear functions. For example, if you are trying to solve the
2
1
2
1
 2  , you can graph f (x) 
equation
and g(x)  2  , and see where they intersect. Look
x2
x
x2
x
at the graph that follows. Based on what you see:
1) Determine the number of solutions for x.
2) Estimate the value of each solution of x.
Now, try this on your own. Use your graphing calculator
to estimate the solutions of each equation. Make sure to
use parentheses correctly when you enter in the equations.
You may need to zoom in/out to get the solutions.
1)
2 x
 8
x 2
2)
x
5
 3
x2
x2
In today’s lesson, we will learn how to solve these equations algebraically.
Pre-AP Algebra 2
8-8 Pair Work
Name: _________________________
Solving Rational Equations
For each equation,
1) determine if you should multiply by the LCD or use cross-multiplication
2) solve the equation
3) determine if any solutions are extraneous
1)
x
4
1
x4
x4
2)
x
2

x 8 x
2
3) 1 
4 12

0
x x2
4)
1
1
10

 2
x3 x3 x 9
5)
2x
x2

4x x4
Pre-AP Algebra 2
Homework #8-8
Name: ________________________
Quiz 2 Review
Do all scratch work on binder paper, stapled to this sheet. Write your answers on this sheet.
Fully simplify each expression. Be sure to indicate values of x that must be excluded from the simplified
expression.
1)
x 3  6x 2  100x  600
x 2  4x  60
2)
2x
6x 2
3x 2  15x
 2
 2
4x  20 x  2x  8
x  16
3
2
2
5x
3)
4
2

10x 10
Perform the indicated operation and simplify.
4)
10
3

2x  3 3  2x
5)
2x  1
4x

x  7x  10 x  5
6)
5
3

x8 x2
2
Solve each equation. Make sure to check your solutions in the original equation and cross out any
extraneous values.
7)
x
5

2x  1 4  x
8)
5x
14
2 2
x 1
x 1
9)
8
3
4


x x5 x
10)
2
1

x  x x 1
2
11)
5x 2  55x  140
.
x 2  5x  14
Make sure to indicate any holes,
intercepts, and asymptotes
clearly on the graph.
Graph f (x) 
Bonus +2.
Homework #8-7 Answer Sheet
1)(
)(
)
y
24
16
8
x
-16
-12
-8
-4
4
8
-8
-16
2)
a)
c)
e)
 x  11
; x  4, 2
( x  4)( x  2)
2 x  13
;x  3
x3
3x  4
; x  2,0, 2
x( x  2)( x  2)
b)
d)
f)
Homework #8-7 Tally Sheet
1)
2)
a)
c)
e)
b)
d)
f)
9 x  27; x  3 / 2,7,5 / 2
4 x2  8x  6
; x  3,3
( x  3)( x  3)( x  3)
17 x
; x  0, 2, 2
3x 2  12