9-4 Simplifying Rational Expressions We simplify rational expressions by cancelling out common factors.
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1 Name Date 9-4 Simplifying Rational Expressions We simplify rational expressions by cancelling out common factors. For example In,both the numerator and denominator have the factor (x+1), so they cancel out, and the simplified answer would be To state restrictions means to tell which x-values would make you divide by zero. To find these, set each factor in the denominator of the original expression equal to zero, and solve. In the example above, (x + 1) = 0 would give you x = -1. Therefore, x -1 is the restriction. Example 1... Simplify each expression. State any restrictions on the variables. (Hint: factor the numerator & denominator first!) c. d.
2 Example 2... Multiply & simplify each expression. State any restrictions on the variables. Example 3... Divide & simplify each expression. State any restrictions on the variables. (Hint: to divide by a fraction means to multiply by the reciprocal so flip the 2 nd fraction!)
3 Name Date 9-5 Adding & Subtracting Rational Expressions When adding or subtracting rational expressions, you have to remember that we re following rules of fractions. And when you add/subtract fractions, you have to have the SAME DENOMINATOR. So, as a simple example Example 1... Find the least common multiple of x 2 9 and x 2 + 6x + 9. Example 2... Add/subtract each rational expression. c. d.
4 A complex fraction is one that has a fraction in the numerator and/or denominator of a bigger fraction. Our goal in simplifying these types of problems is to get rid of the fractions inside the fractions. Example 3... Simplify each complex fraction. a. b.
5 Name Date 9-6 Solving Rational Equations The cross-product method of solving rational equations is used when there is only ONE fraction on either side of an equation. Example 1... Solve each rational equation. If there is more than one fraction on one or both sides of the equation, you must combine the fractions first, THEN use cross products. Example 2... Solve each rational equation. Check your solution.
6 Example 3... One pump can fill a tank with oil in 4 hours. A second pump can fill the same tank in 3 hours. If both pumps are used at the same time, how long will they take to fill the tank? Example 4... Beatrice can travel 40 miles on her motorbike in the same time it takes Tobias to travel 15 miles on his bicycle. If Tobias rides his bike 20 mph slower than Beatrice rides her motorbike, find the speed for each bike.
7 Name Date Notes: Rational Equation Word Problems Rational Equations are used often in problems relating to rate & time or work problems. These problems typically involve two people or machines completing a certain task. We can use the following basic setup to solve these types of problems: Example 1... If two inlet pipes are both open, they can fill a pool in 1 hour and 12 minutes. One of the pipes can fill the pool by itself in 2 hours. How long would it take the other pipe to fill the pool by itself? Example 2... Two Algebra 2 students, Frank and Jill, decide to do their review together to save time. If Frank could finish his review in 3 hours, and Jill could do hers in 2 hours and 15 minutes, how long would it take them to do the review if they work together? Example 3... You and your friend work together doing lawn work for your neighbors. Usually you work together, and it takes an hour and a half to mow & trim one of your neighbor s lawn. Your friend has done it by himself before, and it takes him 2 hours to do by himself. How long would it take you to mow and trim the lawn by yourself?
8 Another type of problem that uses rational expressions are distance/rate/time problems. For these types of problems, it can help to use a table to hold all our information. Just remember that distance always equals rate times time. Example 4... To travel 60 miles, it takes Sue, riding a moped, 2 hours less time than it takes Doreen to travel 50 miles riding a bicycle. Sue travels 10 miles per hour faster than Doreen. Find the rates and times of both girls. distance rate time Sue Doreen Example 5... Julian s boat will go 15 miles per hour in still water. If he can go 12 miles downstream in the same amount of time as it takes him to go 9 miles upstream, then what is the speed of the current? distance rate time Downstream Upstream
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