2. A rational expression is a fraction with polynomials for the numerator and denominator. are rational expressions. For example, If x is replaced by a number making the denominator of a rational expression zero, the value of the rational expression is undefined. Example: Evaluate for x = –3, 0, and 1. x undefined11  3 3 0 0 9

3. A rational equation is an equation between rational expressions. For example, and are rational equations. To solve a rational equation: 1. Factor the denominators to identify the LCM. 2. Multiply all terms by the LCM. 3. Simplify, clearing the Denominators 4. Solve the polynomial. 5. Check your solutions.

4. Example: 1. Solve: Find the LCM. Multiply by LCM = (x – 3). Solve for x. LCM = x – 3. 1 = x + 1 x = 0 Simplify. (0) True. Check. Substitute 0. Simplify.

5. 2. Solve: x – 1 = 2x Find the LCM. LCM = x(x – 1). Multiply all terms by LCM. Simplify. x = –1 Solve. Check.

6. 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).

7. 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.

10. Example 3:

12. Example 5:

13.  CW: WS 5.5  HW:  Pg 346: #33-38 graph (6 steps)  Pg 353: #2-10, 19-27