5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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Bellwork 12-9-14Find the least common multiple for each pair.

1. 2x2 and 4x2 – 2x 2. x + 5 and x2 – x – 30

3. 4. 1 x2

1x –

Add or subtract. Identify any x-values for which the expression is undefined.

1 x – 2

14x +

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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Solve rational equations and inequalities.

Objective

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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rational equationextraneous solutionrational inequality

Vocabulary

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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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.

dr

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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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.

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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December 09, 2014

Solve the equation x – = 3.

Example 1: Solving Rational Equations18x

Multiply each term by the LCD, x.Simplify. Note that x ≠ 0.

Write in standard form.

Factor.

Apply the Zero Product Property.

Solve for x.

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Example 1 Continued

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Check It Out! Example 1a

Multiply each term by the LCD, 3x.

Simplify. Note that x ≠ 0.

Combine like terms.

Solve for x.

Solve the equation = + 2. 4x

103

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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.

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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Solve each equation. Example 2A: Extraneous Solutions

The solution x = 2 is extraneous because it makes the denominators of the original equation equal to 0. Therefore, the equation has no solution.

Divide out common factors.

Multiply each term by the LCD, x – 2.

Simplify. Note that x ≠ 2.

5x x – 2

3x + 4 x – 2 =

Solve for x.

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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December 09, 2014

Check Substitute 2 for x in the original equation.

5x x – 2

3x + 4 x – 2 =

Division by 0 is undefined.

Example 2A Continued

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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Solve each equation.

Example 2B: Extraneous Solutions

Divide out common factors.Multiply each term by the LCD, 2(x – 8).

Simplify. Note that x ≠ 8.Use the Distributive Property.

2x – 5 x – 8

11 x – 8 + =x

2

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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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.

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.

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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Example 6: Solving Rational Inequalities Algebraically

Solve ≤ 3 algebraically. 6x – 8

Case 1 LCD is positive.

Step 1 Solve for x.

6 x – 8 (x – 8) ≤ 3(x – 8) Multiply by the LCD.

Simplify. Note that x ≠ 8.

Solve for x.

Rewrite with the variable on the left.

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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Step 2 Consider the sign of the LCD.

LCD is positive.

Solve for x. ≥ 10 and x > 8, which

Example 6 ContinuedSolve ≤ 3 algebraically. 6

x – 8

For Case 1, the solution must satisfy x ≥ 10 and x > 8, which simplifies to x ≥ 10.

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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Example 6: Solving Rational Inequalities Algebraically

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.

Simplify. Note that x ≠ 8.Solve for x.

Rewrite with the variable on the left.

Solve ≤ 3 algebraically. 6x – 8

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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Step 2 Consider the sign of the LCD.

LCD is positive.Solve for x.

Example 6 Continued

Solve ≤ 3 algebraically. 6x – 8

5­5 Solving Rational Equations and Inequalities­ period 1.notebook

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Homework: 5-5 Worksheet Practice A