rational equations and inequalities

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Holt McDougal Algebra 2 Solving Rational Equations and Inequalities Solve rational equations and inequalities. Objective

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Page 1: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Solve rational equations and inequalities.

Objective

Page 2: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

rational equationextraneous solutionrational inequality

Vocabulary

Page 3: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

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.

dr

Page 4: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

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.

Page 5: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Solve the equation x – = 3.

Example 1: Solving Rational Equations

18x

Page 6: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

x – = 3 18xCheck

1866 – 3

33

6 – 33

x – = 3

18(–3)(–3) – 3

33

–3 + 63

18x

Example 1 Continued

Page 7: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 1a

Solve the equation = + 2. 4x

103

Page 8: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 1b

Solve the equation + = – . 54

6x

74

Page 9: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 1c

Solve the equation x = – 1. 6x

Page 10: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

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.

Page 11: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Solve each equation. Example 2A: Extraneous Solutions

5x x – 2

3x + 4 x – 2 =

Page 12: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check Substitute 2 for x in the original equation.

5x x – 2

3x + 4 x – 2 =

5(2) 2 – 2

3(2) + 4 2 – 2

100

100

Division by 0 is undefined.

Example 2A Continued

Page 13: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Solve each equation. Example 2B: Extraneous Solutions

2x – 5 x – 8

11 x – 8 + =x

2

Page 14: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

2x – 5 x – 8

11 x – 8 + =x

2 2x – 5

x – 8 11 x – 8 + – = 0. x

2

CheckWrite

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.

Example 2B Continued

Page 15: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 2a Solve the equation . 16

x2 – 16 2

x – 4 =

Page 16: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

x x – 1 Solve the equation . 1

x – 1 = + x6

Check It Out! Example 2b

Page 17: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

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.

Page 18: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Example 5: Using Graphs and Tables to Solve Rational Equations and Inequalities

Solve ≤ 3 by using a graph and a table. xx – 6

Page 19: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Example 5: Using Graphs and Tables to Solve Rational Equations and Inequalities

Solve ≤ 3 by using a graph and a table. xx – 6

xx – 6

Use a graph. On a graphing calculator, Y1 = and Y2 = 3.

The graph of Y1 is at or below the graph of Y2 when x < 6 or when x ≥ 9.

(9, 3)

Vertical asymptote:

x = 6

Page 20: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

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.

Page 21: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 5a Solve ≤ 4 by using a graph and a table. x

x – 3

Page 22: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 5a Solve ≤ 4 by using a graph and a table. x

x – 3

xx – 3

Use a graph. On a graphing calculator, Y1 = and Y2 = 4.

The graph of Y1 is at or below the graph of Y2 when x < 3 or when x ≥ 4.

(4, 4)

Vertical asymptote:

x = 3

Page 23: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

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.

Page 24: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 5b

Solve = –2 by using a graph and a table. 8x + 1

Page 25: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 5b

Solve = –2 by using a graph and a table. 8x + 1

The graph of Y1 is at or below the graph of Y2 when x = –5.

(–5, –2)

Vertical asymptote:

x = –1

8x + 1

Use a graph. On a graphing calculator, Y1 = and Y2 = –2.

Page 26: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

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.

Check It Out! Example 5b continued

Page 27: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

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.

Page 28: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Example 6: Solving Rational Inequalities Algebraically

Solve ≤ 3 algebraically. 6x – 8

Case 1 LCD is positive.

Page 29: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Example 6: Solving Rational Inequalities Algebraically

Solve ≤ 3 algebraically. 6x – 8

Case 2 LCD is negative.

Page 30: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 6a

Solve ≥ –4 algebraically. 6x – 2

Case 1 LCD is positive.

Page 31: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 6a Continued

Solve ≥ –4 algebraically. 6x – 2

Case 2 LCD is negative.

Page 32: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Case 1 LCD is positive.

Solve < 6 algebraically. 9x + 3

Check It Out! Example 6b

Page 33: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Check It Out! Example 6b Continued

Case 2 LCD is negative.

Solve < 6 algebraically. 9x + 3

Page 34: Rational Equations and Inequalities

Holt McDougal Algebra 2

Solving Rational Equationsand Inequalities

Lesson Quiz

2.

1. x + 2 x

x – 1 2 =

6xx + 4

7x + 4 x + 4 =

3.

4.

x + 2 x – 3

5 x – 3 + =x

5

Solve each equation or inequality.

4x – 3

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%?

≥ 2