Example 1 Solve a Rational Equation The LCD for the terms is 24(3 – x). Original equation Solve. Check your solution. Multiply each side by 24(3 – x).

Example 1 Solve a Rational Equation Distributive Property Simplify. Add 6x and –63 to each side. Answer:The solution is –45.

Example 2 Solve a Rational Equation The LCD is (p + 1)(p – 1). Original equation Solve Check your solution. Multiply by the LCD.

Example 2 Solve a Rational Equation (p – 1)(p 2 – p – 5) = (p 2 – 7)(p + 1) + p(p + 1)(p – 1) p 3 – p 2 – 5p – p 2 + p + 5 = p 3 + p 2 – 7p – 7 + p 3 – p p 3 – 2p 2 – 4p + 5 = 2p 3 + p 2 – 8p – 7 0= p 3 + 3p 2 – 4p – 12 Divide common factors. Distributive Property Simplify. Subtract p 3 – 2p 2 – 4p + 5 from each side.

Example 2 Solve a Rational Equation Zero Product Property 0=(p + 3)(p + 2)(p – 2) Factor. 0=p + 3 or 0 = p + 2 or 0 = p – 2 Answer: The solutions are –3, –2 and 2.

Example 3 Mixture Problem BRINE Aaron adds an 80% brine (salt and water) solution to 16 ounces of solution that is 10% brine. How much of the solution should be added to create a solution that is 50% brine? UnderstandAaron needs to know how much of a solution needs to be added to an original solution to create a new solution.

Example 3 Mixture Problem PlanEach solution has a certain percentage that is salt. The percentage of brine in the final solution must equal the amount of brine divided by the total solution. Percentage of brine in solution

Example 3 Mixture Problem Substitute. Simplify numerator. LCD is 100(16 + x). SolveWrite a proportion.

Example 3 Mixture Problem Distribute. Subtract 50x and 160. Divide each side by 30. Answer: Aaron needs to add ounces of 80% brine solution. Simplify. Divide common factors.

Example 4 Distance Problem SWIMMING Lilia swims for 5 hours in a stream that has a current of 1 mile per hour. She leaves her dock and swims upstream for 2 miles and then back to her dock. What is her swimming speed in still water? UnderstandWe are given the speed of the current, the distance she swims upstream, and the total time. PlanShe swam 2 miles upstream against the current and 2 miles back to the dock with the current. The formula that relates distance, time, and rate is d = rt or

Example 4 Distance Problem Solve Original equation Time going with the currentplus time going against the currentequals total time. 5 Let r equal her speed in still water. Then her speed with the current is r + 1, and her speed against the current is r – 1.

Example 4 Distance Problem Divide Common Factors Distribute. Simplify. Subtract 4r from each side. (r + 1)2 + (r – 1)2 = 5(r 2 – 1) Simplify. Multiply each side by r 2 – 1.

Example 4 Distance Problem Use the Quadratic Formula to solve for r. Quadratic Formula x = r, a = 5, b = – 4, and c = –5 Simplify.