This document discusses rational expressions, equations, and inequalities. It defines rational expressions as fractions where the numerator and denominator are polynomials, with the denominator not equal to 0. Examples of rational expressions are provided. Methods for simplifying, adding, subtracting, and multiplying rational expressions are outlined. A rational equation is defined as an equation containing one or more rational expressions. Solving rational equations involves eliminating rational expressions by multiplying both sides by the least common denominator. Steps for solving rational equations are demonstrated through examples. Exercises involving solving rational equations are provided at the end for students to practice.

1. Rational Expressions and
Equations
General Mathematics

2. Lesson Objectives
At the end of the lesson, the students must be
able to:
• distinguish rational function, rational
equation, and rational inequality;
• represent real-life situations using rational
functions; and
• solve rational equations and inequalities.

3. Rational Expression
A rational expression can be written in the form
where A and B are polynomials and B ≠ 0.
Examples
A
B
7
ab
x2
- x -6
x +2
x2
+ 7x +10
x +5

6. Simplifying Rational Expression
1. Factor the numerator and denominator.
2. Write a product of two rational expressions,
one factor containing the GCF of the
numerator and denominator, and the other
containing the remaining factors.
3. Rewrite the factor containing the GCF as 1.
4. Multiply the remaining factors by 1.

7. Example 1
Simplify .
Solution
a2
+ 2a
2a+ 4
a2
+2a
2a+ 4
=
a a+ 2
( )
2 a+ 2
( )
=
a
2

8. Adding and Subtracting Rational Expression
with Like Denominators
1. Add (or subtract) the numerators.
2. Retain the common denominator.
3. Simplify the result..
Adding and Subtracting Rational Expression
with Different Denominators
1. Find the least common denominator (LCD).
2. Write the equivalent expression of each rational expression.
3. Add or subtract the numerators and keep the LCD.
4. Simplify the result, if possible..

9. Example 2A
Find the sum of and .
Solution
2a
4b
3
4b
2a
4b
+
3
4b
=
2a+3
4b

10. Example 2B
Add and .
Solution
8d -3
9
4d +12
9
8d -3
9
+
4d +12
9
=
8d -3+ 4d +12
9
=
12d + 9
9
=
3 4d +3
( )
9
=
4d +3
3

11. Multiplying Rational Expression
1. Multiply the numerators and denominators.
2. Write the resulting numerator and denominator in factored
form.
3. Divide out any numerator factor with matching denominator
factor.
4. Simplify, if possible.

12. Example 3
Multiply and .
Solution
a5
10
5
a3
a5
10
×
5
a3
=
5a5
10a3
=
5a3
×a2
5a3
×2
=
a2
2

13. Rational Equation
A rational equation is an equation that
contains one or more rational expressions.
Example
1
x
=
1
5- x
x +
4
x
= -5
x2
x +1
=
1
x +1

14. Solving Rational Equation
1. Eliminate the rational expressions in the
equation by multiplying both sides of the
equation by the LCD.
2. Solve the equation.
3. Check your solution.

15. Example 4A
Solve for x.
Solution
x
5
+
1
4
=
x
2
20
x
5
+
1
4
æ
è
ç
ö
ø
÷ = 20
x
2
æ
è
ç
ö
ø
÷
4x + 5 =10x
5 = 6x
5
6
= x

16. Example 4B
Solve for x.
Solution
1
4
=
3
x
-
1
2
1
4
=
3
x
-
1
2
4x
1
4
æ
è
ç
ö
ø
÷ = 4x
3
x
-
1
2
æ
è
ç
ö
ø
÷
x =12 - 2x
3x =12
x = 4

17. Example 4C
Solve for x.
Solution 4x +1
x +1
-3=
12
x2
-1
x +1
( ) x -1
( )
4x +1
x +1
-3
æ
è
ç
ö
ø
÷ = x +1
( ) x -1
( )
12
x +1
( ) x -1
( )
æ
è
ç
ç
ö
ø
÷
÷
4x +1
( ) x -1
( )-3 x +1
( ) x -1
( ) =12
4x2
-3x -1-3x2
+3=12
x2
-3x -10 = 0
x - 5
( ) x + 2
( ) = 0
x = 5
x = -2
4x +1
x +1
-3=
12
x2
-1

18. Exercises
Solve for x.
1. 4.
2. 5.
3.
y
9
-
2
5
=
1
3
x
5
=15+
x
3
x
6
+
3x
5
= 2
x +1
x - 5
=
5
3
a+2
3
=
a-1
4