1. Find the least common multiple for each pair.
Warm Up
Find the least common multiple for each pair.
1. 2x2 and 4x2 – 2x
2. x + 5 and x2 – x – 30
Add or subtract. Identify any x-values for which the expression is undefined. 1
x – 2 4x + 3. 1 x2 x – 4.

2. Objective Vocabulary Solve rational equations and inequalities.
extraneous solution
rational inequality

3. A rational equation is an equation that contains one or more rational expressions. The time t in hours that it takes to travel d miles can be determined by using the equation t = , where r is the average rate of speed. This equation is a rational equation. d r
To solve a rational equation, start by multiplying each term of the equation by the least common denominator (LCD) of all of the expressions in the equation. This step eliminates the denominators of the rational expression and results in an equation you can solve by using algebra.

4. Example 1: Solving Rational Equations18 x
Solve the equation x – = 3.

5. An extraneous solution is a solution of an equation derived from an original equation that is not a solution of the original equation. When you solve a rational equation, it is possible to get extraneous solutions. These values should be eliminated from the solution set. Always check your solutions by substituting them into the original equation.

6. Example 2A: Extraneous Solutions
Solve each equation. 5x
x – 2
3x + 4 =
2x – 5
x – 8 11 = x 2

7. Example 3: Problem-Solving Application
A jet travels 3950 mi from Chicago, Illinois, to London, England, and 3950 mi on the return trip. The total flying time is 16.5 h. The return trip takes longer due to winds that generally blow from west to east. If the jet’s average speed with no wind is 485 mi/h, what is the average speed of the wind during the round-trip flight? Round to the nearest mile per hour.

8. Example 4: Work Application
Natalie can finish a 500-piece puzzle in about 8 hours. When Natalie and Renzo work together, they can finish a 500-piece puzzle in about 4.5 hours. About how long will it take Renzo to finish a 500-piece puzzle if he works by himself?

9. A rational inequality is an inequality that contains one or more rational expressions. One way to solve rational inequalities is by using graphs and tables.

10. Vertical asymptote: x = 6
Example 5: Using Graphs and Tables to Solve Rational Equations and Inequalities
Solve ≤ 3 by using a graph and a table. x
x – 6
(9, 3)
Vertical asymptote: x = 6

11. You can also solve rational inequalities algebraically
You can also solve rational inequalities algebraically. You start by multiplying each term by the least common denominator (LCD) of all the expressions in the inequality. However, you must consider two cases: the LCD is positive or the LCD is negative.

12. Example 6: Solving Rational Inequalities Algebraically
Solve ≤ 3 algebraically. 6
x – 8
Pg e, 29,30,32-37, 38-46e, 47-49, 54-56, 58-61