How to Solve Rational Equations

1. Solving
Rational Equations
By: Ms. Ruth Good
Algebra 2

2. Rational Equations
• Def: Equations with a variable in
the denominator
• Example:
1
2
1
=
+
+
x
x
x

3. Steps for Solving:
Find LCD of all terms in equation.
Multiply both sides of equation by
LCD.
(This results in a new equation which may
not equal the original equation.)
Solve resulting equation.
Check answers in original
equation.

4. Definition
• Some answers will
not work in the
original equation
because they will
make the
denominator equal
zero. These are
called extraneous
roots.

5. Example 1: Solve
Find LCD. LCD = 2x
xx
12
2
13
=−
x
xx
x 2
12
2
13
2 





=





− Multiply both sides by 2x.
246 =− x
Solve.
18
18
−=
=−
x
x
Now we must check…

6. Checking Ex. 1:
Substitute in x = -18.
18
12
2
1
18
3
−
=−
−
Get common denominators.
18
12
18
12
18
12
18
9
18
3
−
=
−
−
=−
−
It checks!!! X = -18

7. Example 2: Solve
LCD = x + 1
1
5
4
1
5
+
−=
+ xx
x
( ) ( )1
1
5
4
1
5
1 +





+
−=





+
+ x
xx
x
x
5445 −+= xx
1−=x
Solve.
0
5
4
0
5
−=
− Doesn’t check.
No Solution!!
Check:

8. Example 3: Solve
Factor denominator of second fraction.1
4
6
2
23
2
+
−
=
−
−
xx
x
1
)2)(2(
6
2
23
+
−+
=
−
−
xxx
x
LCD = (x+2)(x-2)
)2)(2(1
)2)(2(
6
2
23
)2)(2( −+





+
−+
=





−
−
−+ xx
xxx
x
xx
( )( ) )2)(2(6223 +−+=+− xxxx
1,3
0)1)(3(2
0)32(2
0642
464263
2
2
22
−=
=−+
=−+
=−+
−+=−−+
x
xx
xx
xx
xxxx
Solve.
We still need to check.
X = -3, 1

9. Example 4: Solve
4
1
4
3
2
+
=
+ xxx
What is different about this equation?
3x + 12 = x2
+ 4x Cross-Multiply
0 = x2
+ x – 12 Solve
0 = (x + 4)(x – 3)
x = -4, 3 Check
X = 3

10. Example 4: Solve
4
1
4
3
2
+
=
+ xxx
What is different about this equation?
3x + 12 = x2
+ 4x Cross-Multiply
0 = x2
+ x – 12 Solve
0 = (x + 4)(x – 3)
x = -4, 3 Check
X = 3