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1. Rational Equations and
Inequalities

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.
Rational expressions are also called algebraic
fractions.
Examples

4. The following are not rational expressions:

5. 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.
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6. Example 1
Simplify .
Solution

7. Example 1
Simplify .
Solution

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

10. Example 2B
Add and .
Solution

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

13. Rational Equation
A rational equation is an equation that
contains one or more rational expressions.
Example

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

16. Example 4B
Solve for x.
Solution

17. Example 4C
Solve for x.
Solution

18. Exercises
Solve for x.
1. 4.
2. 5.
3.