1. Solving Rational Equations

2. A rational expression is a fraction with polynomials for the numerator and denominator.
are rational expressions.
For example,
Rational Expression

3. A rational equation is an equation between rational expressions.
For example, and are rational equations.
We are never allowed to divide by zero, so we need to exclude values that make the denominator zero.
To do this we take each denominator that contains a variable and say that it should never be zero. We then solve this to get the excluded values also known as extraneous solutions
Example:
Rational Equation

4. There are two ways to solve a rational equation.
Both ways will give you the same answer and you can choose which one you prefer
The first way involves multiplying each term by the LCD to get rid of the denominators.
The second way involves making all the denominators into the LCD and then “getting rid” of the denominator
I will show you both and you decide which one you prefer.
Both ways require you to find extraneous solutions first!
Rational Equation

5. To solve a rational equation: 1. Find the extraneous solutions.
First way.
To solve a rational equation:
1. Find the extraneous solutions.
2. Find the LCD of the denominators.
3. Clear denominators by multiplying each term on both sides of the equation by the LCD.
4. Solve the resulting polynomial equation.
5. Check the solutions against the extraneous solutions.
Rational Equation

6. To solve a rational equation: 1. Find the extraneous solutions.
Second way.
To solve a rational equation:
1. Find the extraneous solutions.
2. Find the LCD of the denominators.
3. Turn each term’s denominator into the LCD.
4. Now that the denominators are equal we can reason logically that the numerators then have to be equal.
5. Check the solutions against the extraneous solutions.
Rational Equation

7. Examples: 1. Solve: First way LCD = x – 3. 1 = x + 1 x = 0
Find the extraneous solutions.
Find the LCD.
LCD = x – 3.
Multiply each term by LCD
1 = x + 1
Solve for x.
x = 0
Check solution against
Extraneous solutions and cross
off if it is the same.
Examples: Solve

8. Examples: 1. Solve: Second way LCD = x – 3. 1 = x + 1 x = 0
Find the extraneous solutions.
Find the LCD.
LCD = x – 3.
Turn each denominator into LCD
1 = x + 1
Numerators have to be equal.
x = 0
Check solution against
Extraneous solutions and cross
off if it is the same.
Examples: Solve

9. Examples: 2. Solve: First way LCD = x(x – 1). x - 1 = 2x -x = 1 x = -1
Find the extraneous solutions.
Find the LCD.
LCD = x(x – 1).
Multiply each term by LCD
x - 1 = 2x
Solve for x.
-x = 1
x = -1
Check solution against
Extraneous solutions and cross
off if it is the same.
Examples: Solve

10. Examples: 2. Solve: Second way LCD = x(x – 1). x - 1 = 2x -x = 1
Find the extraneous solutions.
Find the LCD.
LCD = x(x – 1).
Make each denominator into LCD
If denominators are equal
then numerators are too
x - 1 = 2x
Solve linear equation
-x = 1
Check solution against
Extraneous solutions and cross
off if it is the same.
x = -1
Examples: Solve

11. Examples: 3. Solve: First way LCD =(x+1)(x – 1). 3x +1 = x - 1 2x = -2
Find the extraneous solutions.
Find the LCD.
LCD =(x+1)(x – 1).
Multiply each term by LCD
3x +1 = x - 1
Solve for x.
Check solution against
Extraneous solutions and cross
off if it is the same.
2x = -2
x = -1
Examples: Solve

12. Examples: 3. Solve: Second way LCD =(x+1)(x – 1). 3x +1 = x - 1
Find the extraneous solutions.
Find the LCD.
LCD =(x+1)(x – 1).
Turn each term into LCD
Solve for x.
3x +1 = x - 1
Check solution against
Extraneous solutions and cross
off if it is the same.
2x = -2
x = -1
Examples: Solve

13. Examples: 4. Solve: First way LCD =x(x – 3). 1 = (x+2)(x-3)-x(x+3)
Find the extraneous solutions.
Find the LCD.
LCD =x(x – 3).
1 = (x+2)(x-3)-x(x+3)
Multiply each term by LCD
1 = x2 – x – 6 - x2 - 3x
Solve for x.
7 = -4x
Check solution against
Extraneous solutions and cross
off if it is the same.
= x
Examples: Solve

14. Examples: 4. Solve: Second way LCD =x(x – 3). 1 = (x+2)(x-3)-x(x+3)
Find the extraneous solutions.
Find the LCD.
LCD =x(x – 3).
1 = (x+2)(x-3)-x(x+3)
Multiply each term by LCD
1 = x2 – x – 6 - x2 - 3x
Solve for x.
7 = -4x
Check solution against
Extraneous solutions and cross
off if it is the same.
= x
Examples: Solve

15. First way Example: Solve: . x2 – 8x + 15 = (x – 3)(x – 5)
Factor.
The LCM is (x – 3)(x – 5).
Original Equation.
x(x – 5) = – 6
Polynomial Equation.
x2 – 5x + 6 = 0
Simplify.
(x – 2)(x – 3) = 0
Factor.
x = 2 or x = 3
Example: Solve

16. Second way Example: Solve: . x2 – 8x + 15 = (x – 3)(x – 5)
Factor.
The LCM is (x – 3)(x – 5).
Original Equation.
Polynomial Equation.
x(x – 5) = – 6
Simplify.
x2 – 5x + 6 = 0
Factor.
(x – 2)(x – 3) = 0
x = 2 or x = 3
Example: Solve

17. Second way Example: Solve: . x2 – 5x + 6 = 0 (x – 2)(x – 3) = 0
Polynomial Equation.
(x – 2)(x – 3) = 0
Factor.
Check solutions against
extraneous solutions
x = 2 or x = 3
Example: Solve

18. Your turn:
Solve:
Example: Solve