This document provides steps for solving rational equations: 1) Find the least common denominator (LCD) of all terms in the equation. 2) Multiply both sides of the equation by the LCD. 3) Solve the resulting equation. 4) Check that any solutions satisfy the original equation, as some solutions may be "extraneous roots" that make the denominator equal to zero. The document includes examples demonstrating these steps, such as solving equations with factored denominators and equations where cross-multiplying eliminates the fractions.

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