10 7 Solving Rational Equations Preview Warm Up

10 -7 Solving Rational Equations Preview Warm Up California Standards Lesson Presentation

10 -7 Solving Rational Equations Warm Up 1. Find the LCM of x, 2 x 2, and 6. 1. 2. Find the LCM of p 2 – 4 p and p 2 – 16. 2. Multiply. Simplify your answer. 3. 3. 4. 5. 5. 4.

10 -7 Solving Rational Equations California Standards Preparation for 15. 0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.

10 -7 Solving Rational Equations Vocabulary rational equation extraneous solutions

10 -7 Solving Rational Equations A rational equation is an equation that contains one or more rational expressions. If a rational equation is a proportion, it can be solved using the Cross Product Property.

10 -7 Solving Rational Equations Additional Example 1: Solving Rational Equations by Using Cross Products Solve . Check your answer. 5 x = (x – 2)(3) Use cross products. 5 x = 3 x – 6 Distribute 3 on the right side. 2 x = – 6 Subtract 3 x from both sides. x = – 3 Divide both sides by 2.

10 -7 Solving Rational Equations Additional Example 1 Continued Check Substitute – 3 for x in the original equation. – 1

10 -7 Solving Rational Equations Check It Out! Example 1 a Solve 3 n = (n + 4)(1) . Check your answer. 3 n = n + 4 Use cross products. Distribute 1 on the right side. 2 n = 4 Subtract n from both sides. n=2 Divide both sides by 2.

10 -7 Solving Rational Equations Check It Out! Example 1 a Continued Check Substitute 2 for n in the original equation.

10 -7 Solving Rational Equations Check It Out! Example 1 b Solve . Check your answer. 4 h = (h + 1)(2) Use cross products. 4 h = 2 h + 2 Distribute 2 on the right side. 2 h = 2 Subtract 2 h from both sides. h=1 Divide both sides by 2.

10 -7 Solving Rational Equations Check It Out! Example 1 b Continued Check Substitute 1 for h in the original equation.

10 -7 Solving Rational Equations Check It Out! Example 1 c Solve . Check your answer. 21 x = (x – 7)(3) Use cross products. 21 x = 3 x – 21 Distribute 3 on the right side. 18 x = – 21 Subtract 3 x from both sides. x= Divide both sides by 18.

10 -7 Solving Rational Equations Check It Out! Example 1 c Continued Check Substitute for x in the original equation.

10 -7 Solving Rational Equations Some rational equations contain sums or differences of rational expressions. To solve these, you must find a common denominator for all the rational expressions in the equation.

10 -7 Solving Rational Equations Additional Example 2 A: Solving Rational Equations by Using the LCD Solve each equation. Check your answer. Step 1 Find the LCD. 2 x(x + 1) Include every factor of the denominator. Step 2 Multiply both sides by the LCD. Distribute on the left side.

10 -7 Solving Rational Equations Additional Example 2 A Continued Step 3 Simplify and solve. Divide out common factors. (2 x)(2) + 6(x + 1) = 5(x + 1) Simplify. 4 x + 6 = 5 x + 5 Distribute and multiply. 10 x + 6 = 5 x + 5 Combine like terms. Subtract 5 x and 6 from both sides. Divide both sides by 5. 5 x = – 1

10 -7 Solving Rational Equations Additional Example 2 A Continued Check

10 -7 Solving Rational Equations Additional Example 2 B: Solving Rational Equations by Using the LCD Solve each equation. Check your answer. Step 1 Find the LCD. (x 2) Include every factor of the denominator. Step 2 Multiply both sides by the LCD. Distribute on the left side.

10 -7 Solving Rational Equations Additional Example 2 B Continued Step 3 Simplify and solve. Divide out common factors. 4 x – 3 = x 2 –x 2 + 4 x – 3 = 0 x 2 – 4 x + 3 = 0 (x – 3)(x – 1) = 0 Simplify. Subtract x 2 from both sides. Multiply by – 1. Factor. x = 3 or 1 Solve.

10 -7 Solving Rational Equations Additional Example 2 B Continued Check

10 -7 Solving Rational Equations Check It Out! Example 2 a Solve each equation. Check your answer. Step 1 Find the LCD. a(a +1) Include every factor of the denominator. Step 2 Multiply both sides by the LCD. Distribute on the left side.

10 -7 Solving Rational Equations Check It Out! Example 2 a Continued Step 3 Simplify and solve. Divide out common factors. 3 a = 4(a + 1) Simplify. 3 a = 4 a + 4 Distribute the 4. – 4 = a Subtract the 4 and 3 a from both sides.

10 -7 Solving Rational Equations Check It Out! Example 2 a Continued Check

10 -7 Solving Rational Equations Check It Out! Example 2 b Solve each equation. Check your answer. Step 1 Find the LCD. 2 j(j +2) Include every factor of the denominator. Step 2 Multiply both sides by the LCD. Distribute on the left side.

10 -7 Solving Rational Equations Check It Out! Example 2 b Continued Solve each equation. Check your answer. Divide out common factors. 12 j – 10(2 j + 4) = 4 j + 8 12 j – 20 j – 40 = 4 j + 8 – 12 j = 48 j = – 4 Simplify. Distribute 10. Combine like terms. Divide both sides by – 12.

10 -7 Solving Rational Equations Check It Out! Example 2 b Continued Check

10 -7 Solving Rational Equations When you multiply each side of an equation by the LCD, you may get an extraneous solution. An extraneous solution is a solution to a resulting equation that is not a solution to the original equation. Because of extraneous solutions, it is especially important to check your answers.

10 -7 Solving Rational Equations Additional Example 3: Extraneous Solutions Solve . Check your answer. 2(x 2 – 1) = (x + 1)(x – 6) Use cross products. Distribute 2 on the left side. 2 x 2 – 2 = x 2 – 5 x – 6 Multiply the right side. Subtract x 2 from both sides. x 2 + 5 x + 4 = 0 Add 5 x and 6 to both sides. Factor the quadratic expression. (x + 1)(x + 4) = 0 Use the Zero Product Property. x = – 1 or x = – 4 Solve.

10 -7 Solving Rational Equations Additional Example 3 Continued Solve . Check your answer. Check Because and are undefined, – 1 is not a solution. – 1 is an extraneous solution. The only solution is – 4.

10 -7 Solving Rational Equations Check It Out! Example 3 a Solve. Check your answer. (x – 2)(x – 7) = 3(x – 7) Use cross products. x 2 – 9 x + 14 = 3 x – 21 Distribute 3 on the right side. Multiply the left side. x 2 – 12 x + 35 = 0 Subtract 3 x from both sides. Add 21 to both sides. (x – 7)(x – 5) = 0 x = 7 or x = 5 Factor the quadratic expression. Use the Zero Product Property. Solve.

10 -7 Solving Rational Equations Check It Out! Example 3 a Continued Check Because and are undefined, 7 is not a solution. 7 is an extraneous solution. The only solution is 5.

10 -7 Solving Rational Equations Check It Out! Example 3 b Solve. Check your answer. (x + 1)(x – 3) = 4(x – 2) Use cross products. x 2 – 2 x – 3 = 4 x – 8 Distribute 4 on the right side. Multiply the left side. Subtract 4 x from both sides. x 2 – 6 x + 5 = 0 Add 8 to both sides. (x – 1)(x – 5) = 0 x = 1 or x = 5 Factor the quadratic expression. Use the Zero Product Property. Solve.

10 -7 Solving Rational Equations Check It Out! Example 3 b Continued Check 1 and 5 are solutions. There are no extraneous solutions. The solutions are 1 and 5.

10 -7 Solving Rational Equations Check It Out! Example 3 c Solve. Identify any extraneous solutions. 6(x 2 + 2 x) = 9(x 2) 6 x 2 + 12 x) = 9 x 2 3 x 2 – 12 x = 0 3 x(x – 4) = 0 3 x = 0, or x – 4 = 0 x = 0 or x = 4 Use cross products. Distribute 6 on the left side. Multiply the right side. Subtract 9 x 2 from both sides. Multiply through with – 1. Factor the quadratic expression. Use the Zero Product Property. Solve.

10 -7 Solving Rational Equations Check It Out! Example 3 c Continued Check Because and are undefined, 0 is not a solution. 0 is an extraneous solution. The only solution is 4.

10 -7 Solving Rational Equations Lesson Quiz Solve each equation. Check your answer. 1. 24 – 4, 3 2. 3. 4. Solve Check your answer. – 5 .