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**Solving Rational Equations and Inequalities**

8-5 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

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**Find the least common multiple for each pair.**

Warm Up Find the least common multiple for each pair. 1. 2x2 and 4x2 – 2x 2x2(2x – 1) 2. x + 5 and x2 – x – 30 (x + 5)(x – 6) Add or subtract. Identify any x-values for which the expression is undefined. 1 x – 2 4x + 3. 5x – 2 4x(x – 2) –(x – 1) x2 1 x2 x – x ≠ 0 4.

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Objective Solve rational equations and inequalities.

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Vocabulary rational equation extraneous solution rational inequality

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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

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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.

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**Example 1: Solving Rational Equations**

18 x Solve the equation x – = 3. x(x) – (x) = 3(x) 18 x Multiply each term by the LCD, x. x2 – 18 = 3x Simplify. Note that x ≠ 0. x2 – 3x – 18 = 0 Write in standard form. (x – 6)(x + 3) = 0 Factor. x – 6 = 0 or x + 3 = 0 Apply the Zero Product Property. x = 6 or x = –3 Solve for x.

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Example 1 Continued 18 x 18 x Check x – = 3 x – = 3 18 6 6 – 18 (–3) (–3) – 3 3 6 – 3 3 –3 + 6 3 3 3 3 3

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Check It Out! Example 1a Solve the equation = 4 x 10 3 (3x) = (3x) + 2(3x) 10 3 4 x Multiply each term by the LCD, 3x. 10x = x Simplify. Note that x ≠ 0. 4x = 12 Combine like terms. x = 3 Solve for x.

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Check It Out! Example 1b Solve the equation = – . 5 4 6 x 7 (4x) (4x) = – (4x) 6 x 5 4 7 Multiply each term by the LCD, 4x. 24 + 5x = –7x Simplify. Note that x ≠ 0. 24 = –12x Combine like terms. x = –2 Solve for x.

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Check It Out! Example 1c 6 x Solve the equation x = – 1. x(x) = (x) – 1(x) 6 x Multiply each term by the LCD, x. x2 = 6 – x Simplify. Note that x ≠ 0. x2 + x – 6 = 0 Write in standard form. (x – 2)(x + 3) = 0 Factor. x – 2 = 0 or x + 3 = 0 Apply the Zero Product Property. x = 2 or x = –3 Solve for x.

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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.

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**Example 2A: Extraneous Solutions**

Solve each equation. 5x x – 2 3x + 4 = 5x x – 2 3x + 4 (x – 2) = (x – 2) Multiply each term by the LCD, x – 2. 5x x – 2 3x + 4 (x – 2) = (x – 2) Divide out common factors. 5x = 3x + 4 Simplify. Note that x ≠ 2. x = 2 Solve for x. The solution x = 2 is extraneous because it makes the denominators of the original equation equal to 0. Therefore, the equation has no solution.

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Example 2A Continued Check Substitute 2 for x in the original equation. 5x x – 2 3x + 4 = 5(2) 2 – 2 3(2) + 4 2 – 2 10 10 Division by 0 is undefined.

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**Example 2B: Extraneous Solutions**

Solve each equation. 2x – 5 x – 8 x 2 11 x – 8 = Multiply each term by the LCD, 2(x – 8). 2x – 5 x – 8 2(x – 8) (x – 8) = (x – 8) 11 x 2 Divide out common factors. 2x – 5 x – 8 2(x – 8) (x – 8) = (x – 8) 11 x 2 2(2x – 5) + x(x – 8) = 11(2) Simplify. Note that x ≠ 8. Use the Distributive Property. 4x – 10 + x2 – 8x = 22

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Example 2B Continued x2 – 4x – 32 = 0 Write in standard form. (x – 8)(x + 4) = 0 Factor. x – 8 = 0 or x + 4 = 0 Apply the Zero Product Property. x = 8 or x = –4 Solve for x. The solution x = 8 us extraneous because it makes the denominator of the original equation equal to 0. The only solution is x = –4.

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Example 2B Continued Check Write 2x – 5 x – 8 11 = x 2 2x – 5 x – 8 11 – = 0. x 2 as Graph the left side of the equation as Y1. Identify the values of x for which Y1 = 0. The graph intersects the x-axis only when x = –4. Therefore, x = –4 is the only solution.

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Check It Out! Example 2a 16 x2 – 16 2 x – 4 Solve the equation = Multiply each term by the LCD, (x – 4)(x +4). 16 (x – 4)(x + 4) 2 x – 4 (x – 4)(x + 4) = (x – 4 )(x + 4) Divide out common factors. 16 (x – 4)(x + 4) 2 x – 4 (x – 4)(x + 4) = (x – 4 )(x + 4) 16 = 2x + 8 Simplify. Note that x ≠ ±4. x = 4 Solve for x. The solution x = 4 is extraneous because it makes the denominators of the original equation equal to 0. Therefore, the equation has no solution.

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Check It Out! Example 2b x x – 1 Solve the equation 1 = 6 Multiply each term by the LCD, 6(x – 1). 1 x – 1 6(x – 1) = (x – 1) (x – 1) x 6 Divide out common factors. 1 x – 1 6(x – 1) = (x – 1) (x – 1) x 6 6 = 6x + x(x – x) Simplify. Note that x ≠ 1. Use the Distributive Property. 6 = 6x + x2 – x

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**Check It Out! Example 2b Continued**

0 = x2 + 5x – 6 Write in standard form. 0 = (x + 6)(x – 1) Factor. x + 6 = 0 or x – 1 = 0 Apply the Zero Product Property. x = –6 or x = 1 Solve for x. The solution x = 1 us extraneous because it makes the denominator of the original equation equal to 0. The only solution is x = –6.

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**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.

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**Understand the Problem**

Example 3 Continued 1 Understand the Problem The answer will be the average speed of the wind. List the important information: The jet spent 16.5 h on the round-trip. It went 3950 mi east and 3950 mi west. Its average speed with no wind is 485 mi/h.

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**Example 3 Continued 2 Make a Plan**

Let w represent the speed of the wind. When the jet is going east, its speed is equal to its speed with no wind plus w. When the jet is going west, its speed is equal to its speed with no wind minus w. Distance (mi) Average Speed (mi/h) Time (h) East 3950 485 + w West 485 – w 485 + w 3950 485 – w 3950 total time = time east + time west 16.5 485 + w 3950 485 – w = +

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**Use the Distributive Property.**

Solve 3 The LCD is (485 + w)(485 – w). 16.5(485 + w)(485 – w) 485 + w 3950 = (485 + w)(485 – w) 485 – w (485 + w)(485 – w) Simplify. Note that x ≠ ±485. 16.5(485 + w)(485 – w) = 3950(485 – w) (485 + w) Use the Distributive Property. 3,881,212.5 – 16.5w2 = 1,915,750 – 3950w + 1,915, w 3,881,212.5 – 16.5w2 = 3,831,500 Combine like terms. –16.5w2 = –49,712.5 Solve for w. w ≈ ± 55 The speed of the wind cannot be negative. Therefore, the average speed of the wind is 55 mi/h.

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Example 3 Continued Look Back 4 If the speed of the wind is 55 mi/h, the jet’s speed when going east is = 540 mi/h. It will take the jet approximately 7.3 h to travel 3950 mi east. The jet’s speed when going west is 485 – 55 = 430 mi/h. It will take the jet approximately 9.2 h to travel 3950 mi west. The total trip will take 16.5 h, which is the given time.

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Check It Out! Example 3 On a river, a kayaker travels 2 mi upstream and 2 mi downstream in a total of 5 h. In still water, the kayaker can travel at an average speed of 2 mi/h. Based on this information, what is the average speed of the current of this river? Round to the nearest tenth.

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**Understand the Problem**

Check It Out! Example 3 Continued 1 Understand the Problem The answer will be the average speed of the current. List the important information: The kayaker spent 5 hours kayaking. She went 2 mi upstream and 2 mi downstream. Her average speed in still water is 2 mi/h.

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**Check It Out! Example 3 Continued**

2 Make a Plan Let c represent the speed of the current. When the kayaker is going upstream, her speed is equal to her speed in still water minus c. When the kayaker is going downstream, her speed is equal to her speed in still water plus c. Distance (mi) Average Speed (mi/h) Time (h) Up 2 2 – c Down 2 + c 2 – c 2 2 + c 2 total time = time up- stream + time down- stream 5 2 – c 2 2 + c = +

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**Use the Distributive Property.**

Solve 3 The LCD is (2 – c)(2 + c). (2 + c)(2 – c) = (2 + c)(2 – c) 5(2 + c)(2 – c) 2 – c 2 2 + c Simplify. Note that x ≠ ±2. 5(2 + c)(2 – c) = 2(2 + c) + 2(2 – c) Use the Distributive Property. 20 – 5c2 = 4 + 2c + 4 – 2c 20 – 5c2 = 8 Combine like terms. –5c2 = –12 Solve for c. c ≈ ± 1.5 The speed of the current cannot be negative. Therefore, the average speed of the current is about 1.5 mi/h.

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**Check It Out! Example 3 Continued**

Look Back 4 If the speed of the current is about 1.5 mi/h, the kayaker’s speed when going upstream is 2 – 1.5 = 0.5 mi/h. It will take her about 4 h to travel 2 mi upstream. Her speed when going downstream is about = 3.5 mi/h. It will take her 0.5 h to travel 2 mi downstream. The total trip will take about 4.5 hours which is close to the given time of 5 h.

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**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? 1 8 Natalie’s rate: of the puzzle per hour 1 h Renzo’s rate: of the puzzle per hour, where h is the number of hours needed to finish the puzzle by himself.

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**Example 4 Continued Natalie’s rate hours worked Renzo’s rate**

1 complete puzzle + = 1 8 (4.5) 1 h (4.5) + = 1 1 8 (4.5)(8h) + h (4.5)(8h) = 1(8h) Multiply by the LCD,8h. 4.5h + 36 = 8h Simplify. 36 = 3.5h Solve for h. 10.3 = h It will take Renzo about 10.3 hours, or 10 hours 17 minutes to complete a 500-piece puzzle working by himself.

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Check It Out! Example 4 Julien can mulch a garden in 20 minutes. Together Julien and Remy can mulch the same garden in 11 minutes. How long will it take Remy to mulch the garden when working alone? Julien’s rate: of the garden per minute 1 20 Remy’s rate: of the garden per minute, where m is the number of minutes needed to mulch the garden by himself. 1 m

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**Check It Out! Example 4 Continued**

Julien’s rate min worked Remy’s rate 1 complete job + = 1 20 (11) 1 m (11) + = 1 1 20 (11)(20m)+ m (11)(20m) = 1(20m) Multiply by the LCD, 20m. 11m = 20m Simplify. 220 = 9m Solve for m. 24 ≈ m It will take Remy about 24 minutes to mulch the garden working by himself.

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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.

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**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 x x – 6 Use a graph. On a graphing calculator, Y1 = and Y2 = 3. (9, 3) The graph of Y1 is at or below the graph of Y2 when x < 6 or when x ≥ 9. Vertical asymptote: x = 6

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Example 5 Continued Use a table. The table shows that Y1 is undefined when x = 6 and that Y1 ≤ Y2 when x ≥ 9. The solution of the inequality is x < 6 or x ≥ 9.

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**Vertical asymptote: x = 3**

Check It Out! Example 5a Solve ≥ 4 by using a graph and a table. x x – 3 x x – 3 Use a graph. On a graphing calculator, Y1 = and Y2 = 4. (4, 4) The graph of Y1 is at or below the graph of Y2 when x < 3 or when x ≥ 4. Vertical asymptote: x = 3

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**Check It Out! Example 5a continued**

Use a table. The table shows that Y1 is undefined when x = 3 and that Y1 ≤ Y2 when x ≥ 4. The solution of the inequality is x < 3 or x ≥ 4.

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**Vertical asymptote: x = –1**

Check It Out! Example 5b Solve = –2 by using a graph and a table. 8 x + 1 8 x + 1 Use a graph. On a graphing calculator, Y1 = and Y2 = –2. (–5, –2) The graph of Y1 is at or below the graph of Y2 when x = –5. Vertical asymptote: x = –1

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**Check It Out! Example 5b continued**

Use a table. The table shows that Y1 is undefined when x = –1 and that Y1 ≤ Y2 when x = –5. The solution of the inequality is x = –5.

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**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.

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**Example 6: Solving Rational Inequalities Algebraically**

Solve ≤ 3 algebraically. 6 x – 8 Case 1 LCD is positive. Step 1 Solve for x. 6 x – 8 (x – 8) ≤ 3(x – 8) Multiply by the LCD. 6 ≤ 3x – 24 Simplify. Note that x ≠ 8. 30 ≤ 3x Solve for x. 10 ≤ x Rewrite with the variable on the left. x ≥ 10

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Example 6 Continued Solve ≤ 3 algebraically. 6 x – 8 Step 2 Consider the sign of the LCD. x – 8 > 0 LCD is positive. x > 8 Solve for x. For Case 1, the solution must satisfy x ≥ 10 and x > 8, which simplifies to x ≥ 10.

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**Example 6: Solving Rational Inequalities Algebraically**

Solve ≤ 3 algebraically. 6 x – 8 Case 2 LCD is negative. Step 1 Solve for x. 6 x – 8 (x – 8) ≥ 3(x – 8) Multiply by the LCD. Reverse the inequality. 6 ≥ 3x – 24 Simplify. Note that x ≠ 8. 30 ≥ 3x Solve for x. 10 ≥ x Rewrite with the variable on the left. x ≤ 10

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**Step 2 Consider the sign of the LCD.**

Example 6 Continued Solve ≤ 3 algebraically. 6 x – 8 Step 2 Consider the sign of the LCD. x – 8 > 0 LCD is positive. x > 8 Solve for x. For Case 2, the solution must satisfy x ≤ 10 and x < 8, which simplifies to x < 8. The solution set of the original inequality is the union of the solutions to both Case 1 and Case 2. The solution to the inequality ≤ 3 is x < 8 or x ≥ 10, or {x|x < 8 x ≥ 10}. 6 x – 8

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Check It Out! Example 6a Solve ≥ –4 algebraically. 6 x – 2 Case 1 LCD is positive. Step 1 Solve for x. 6 x – 2 (x – 2) ≥ –4(x – 2) Multiply by the LCD. 6 ≥ –4x + 8 Simplify. Note that x ≠ 2. –2 ≥ –4x Solve for x. ≤ x 1 2 Rewrite with the variable on the left. x ≥ 1 2

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**Check It Out! Example 6a Continued**

Solve ≥ –4 algebraically. 6 x – 2 Step 2 Consider the sign of the LCD. x – 2 > 0 LCD is positive. x > 2 Solve for x. For Case 1, the solution must satisfy and x > 2, which simplifies to x > 2. x ≥ 1 2

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**Check It Out! Example 6a Continued**

Solve ≥ –4 algebraically. 6 x – 2 Case 2 LCD is negative. Step 1 Solve for x. 6 x – 2 (x – 2) ≤ –4(x – 2) Multiply by the LCD. Reverse the inequality. 6 ≤ –4x + 8 Simplify. Note that x ≠ 2. –2 ≤ –4x Solve for x. ≥ x 1 2 Rewrite with the variable on the left. x ≤ 1 2

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**Check It Out! Example 6a Continued**

Solve ≥ –4 algebraically. 6 x – 2 Step 2 Consider the sign of the LCD. x – 2 < 0 LCD is negative. x < 2 Solve for x. For Case 2, the solution must satisfy and x < 2, which simplifies to x ≤ 1 2 The solution set of the original inequality is the union of the solutions to both Case 1 and Case 2. The solution to the inequality ≥ –4 is x > 2 or x ≤ , or {x| x > 2}. 6 x – 2 x ≤ 1 2

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Check It Out! Example 6b Solve < 6 algebraically. 9 x + 3 Case 1 LCD is positive. Step 1 Solve for x. 9 x + 3 (x + 3) < 6(x + 3) Multiply by the LCD. 9 < 6x + 18 Simplify. Note that x ≠ –3. –9 < 6x Solve for x. – < x 3 2 Rewrite with the variable on the left. x > – 3 2

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**Check It Out! Example 6b Continued**

Solve < 6 algebraically. 9 x + 3 Step 2 Consider the sign of the LCD. x + 3 > 0 LCD is positive. x > –3 Solve for x. For Case 1, the solution must satisfy and x > –3, which simplifies to x >– 3 2

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**Check It Out! Example 6b Continued**

Solve < 6 algebraically. 9 x + 3 Case 2 LCD is negative. Step 1 Solve for x. Multiply by the LCD. Reverse the inequality. 9 x + 3 (x + 3) > 6(x + 3) 9 > 6x + 18 Simplify. Note that x ≠ –3. –9 > 6x Solve for x. – > x 3 2 Rewrite with the variable on the left. x < – 3 2

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**Check It Out! Example 6b Continued**

Solve < 6 algebraically. 9 x + 3 Step 2 Consider the sign of the LCD. x + 3 < 0 LCD is negative. x < –3 Solve for x. For Case 2, the solution must satisfy and x < –3, which simplifies to x < –3. x <– 3 2 The solution set of the original inequality is the union of the solutions to both Case 1 and Case 2. The solution to the inequality < 6 is x < –3 or x > – , or {x| x < –3}. x > – 3 2 9 x + 3

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Lesson Quiz Solve each equation or inequality. x + 2 x x – 1 2 = 1. x = –1 or x = 4 6x x + 4 7x + 4 = 2. no solution 3. x + 2 x – 3 x 5 5 x – 3 = x = –5 4 x – 3 4. ≥ 2 3 < x ≤ 5 5. A college basketball player has made 58 out of 82 attempted free-throws this season. How many additional free-throws must she make in a row to raise her free-throw percentage to 90%? 158

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