Download Rational Equations

Objectives:
1. Be able to determine if an equation is a
rational equation.
2. Be able to solve various rational equations
and exclude any extraneous solutions.
Critical Vocabulary:
Rational Function, Extraneous solution
Rational equation
•An equation that contains one or more rational
expressions
•Solve a rational equation by multiplying both sides of
the equal sign by the LCD
•If both sides of the equal sign are rational, you can
solve be cross multiplying because it will be a
proportion
•Least common multiple: smallest positive whole
number exactly divisible by the given numbers
I. Rational Functions
A rational equation is an equation that contains rational
expressions (x in the denominator)
Formal Definition: A rational Function is a ratio of two
p ( x)
polynomials written in the form f ( x) 
q ( x)
2
3x  4 x  7
f ( x) 
2x  8
No,
Looks
that
like
would
you
be got
a linear
it.
equation.
In Simple terms,
it’s a fraction.
I think I get it. I
What
a Rational
betisthis
is a
Equation?
1 equation.
2
rational
x x 7
5
3
So,
this
So,
what
a
5 would
3 is
 then?
7be
equation
rational
x x
rational?
II. Solving Rational Equations
b. Solving a Rational: By Finding LCM (Denominator)
1.
5
1
1


12 2 x 3 x
What can x not be?
x0
12x  5  1  1 


 12 2 x 3 x 
Multiply by LCD
Distribute
5x –6 =4
5x = 10
Solution: x = 2
No….really?
This is not an
extraneous
solution either.
II. Solving Rational Equations
a. Solving a Rational: Cross Multiplication (Proportion)
5.
6
x3

x5 x5
First determine
what “x” can’t be
x  5
(6)(x + 5) = (x +5)(x + 3)
Cross Multiply
6x+30 = x2 + 8x + 15
Make it equal 0
0 = x2 +2x-15
0= (x + 5)(x – 3)
x = -5 x = 3
Solution: x = 3
What are talking about?
What is an extraneous
solution?
-5
That’s
is anwhere
extraneous
your solution
solution.
is one of the values that “x”
can’t be.
II. Solving Rational Equations
b. Solving a Rational: By Finding LCM (Denominator)
9.
5 6
 1
2
x
x
What can x not be?
2
6 
 5
 2   1
x 
x
Multiply by LCD
x
Distribute
5 =6x -x2
Make it equal 0
x2 –6x+5 =0
(x - 1)(x - 5)=0
x=1
right
Solution: x = 1,5
x0
x=5
Both of these
solutions look
good.