1. Algebraic Fractions and Rational Equations

2. In this discussion, we will look at examples of simplifying Algebraic Fractions using the 4 rules of fractions.

3. Addition Simplify

4. Addition

5. Subtraction

6. Subtraction 4

7. Multiplication

8. Multiplication Factoring (2x-6) 3

9. Division Simplify 2 2

10. Equations Once you know how to simplify algebraic fractions, you Can solve equations containing them For example Click an equation to see it solved

11. Solve

13. Rational Equation The equation in the previous example is called a rational equation. and are rational equations. 4. Check the solutions. 3. Solve the resulting polynomial equation. 2. Clear denominators by multiplying both sides of the equation by the LCM. 1. Find the LCM of the denominators. To solve a rational equation:

14. Examples: Solve Examples: 1. Solve:. Multiply by LCM = (x – 3). Solve for x. 1 = x + 1 x = 0 Check. Simplify. (0) True. 2. Solve:. x – 1 = 2x Find the LCM. LCM = x(x – 1). Multiply by LCM. Simplify. x = –1 Solve.

15. Example: Solve After clearing denominators, a solution of the polynomial equation may make a denominator of the rational equation zero. Since x 2 – 1 = (x – 1)(x + 1), Since – 1 makes both denominators zero, the rational equation has no solutions. Example: Solve:. 2x = – 2  x = – 1 3x + 1 = x – 1 Check. It is critical to check all solutions. In this case, the value is not a solution of the rational equation. LCM = (x – 1)(x + 1).

16. Example: Solve 16 Example: Solve:. Factor. Polynomial Equation. Simplify. Factor. The LCM is (x – 3)(x – 5). x 2 – 8x + 15 = (x – 3)(x – 5) x(x – 5) = – 6 x 2 – 5x + 6 = 0 (x – 2)(x – 3) = 0 x = 2 or x = 3 Check. x = 2 is a solution. Check. x = 3 is not a solution since both sides would be undefined. Original Equation.