Objectives
Find the values that will make a
rational expression
undefined(excluded values)
Solve rational equations
Algebraic Fraction/Rational
Expression
5
y-7
3x + 2
2
x -4
2
5rs
3t + t + 5
4
t +1
2
Since a RE indicates division, we cannot
replace the variables in a denominator with
numbers that make the denominator zero.
State the value of the variable
that will make the expression
undefined.
2
x
y
4x
State the value of the variable
that will make the expression
undefined.
m
2
y
x-4
State the value of the variable
that will make the expression
undefined.
k -1
2 -k
5w - z
2
x -9
State the value of the variable
that will make the expression
undefined.
x -1
2
2x - 5x - 3
2
Recall: Find the LCD
5
x
10x
,
,
6xy 2x - 4 4 - 2x
Recall: Find the LCD
4 3
x
,
, 2
y 2 + y 2y + 4y
Recall: Find the LCD
d
3
d
,
, 2
3- d d - 3 d - 9
Solving Rational Equations
1. Identify and exclude any values that
will make the denominator zero.
2. Multiply both sides by the LCD and
simplify (this will eliminate all
denominators).
3. Solve the resulting equation.
4. Check all solutions in the original
equation.
Example 12
Solve the equation.
b - 2 b -1
+
=1
5
3
æ b - 2 b -1
ö
+
= 1÷15
ç
è 5
ø
3
3(b - 2) + 5(b -1) = 15
3b - 6 + 5b - 5 = 15
8b -11 = 15
8b = 26
26
1
b=
or3
8
4
Multiply by the LCD: 15
Example 13
Solve the equation.
6
4
=0
2b - 5 b - 3
Example 13
6(b - 3) - 4(2b - 5) = 0
6b -18 - 8b + 20 = 0
-2b + 2 = 0
b =1
Check:
We need 2b – 5 ≠ 0 and b – 3 ≠ 0
which means b ≠ 5/2 and b ≠ 3.
multiply
by LCD.
Simplify – denominators
are eliminated
Distribute
Solve for b.
Solve for m.
2
1
4
= 2
m m -1 m - m
Since m2- m = m(m – 1), the LCD is m(m – 1),
where m≠0 and m ≠ 1.
Solve for m.
æ2
1 ö é 4 ù
m(m -1)ç ÷ = ê 2 ú m(m -1)
è m m -1 ø ë m - m û
2(m – 1) – m = 4
2m - 2 - m = 4
m=6
multiply
by LCD.
Simplify – denominators
are eliminated
Distribute
Solve for m.
Solve for x.
12
4x
x+
= 1+
x -3
x -3
x2 – 3x + 12 = x – 3 + 4x
x2 – 8x + 15 = 0
(x – 3)(x – 5) = 0
x = 3 or x = 5
multiply
by LCD.
Simplify – denominators
are eliminated
Set equation equal to
zero.
Factor.
Zero factor property
x = 3 is an extraneous root and x =5 is the only solution.
SOLVING
RATIONAL EQUATIONS
• Find the LCD
• Multiply both sides of the equation
by the LCD.
• Solve using APE/MPE
and/or
• Solve by factoring.
• Cross check solutions with
restrictions
Solve for x.
12
4x
x+
= 1+
x -3
x -3
Solve for y.
y
4
3+
=
y-4 y-4