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6.6 Solving Equations with Rational Expressions

6.6 Solving Equations with Rational Expressions. Objective 1. Distinguish between operations with rational expressions and equations with terms that are rational expressions. Slide 6.6-3.

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6.6 Solving Equations with Rational Expressions

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  1. 6.6 Solving Equations with Rational Expressions

  2. Objective 1 Distinguish between operations with rational expressions and equations with terms that are rational expressions. Slide 6.6-3

  3. Distinguish between operations with rational expressions and equations with terms that are rational expressions. Before solving equations with rational expressions, you must understand the difference between sums and differences of terms with rational coefficients, or rational expressions, and equations with terms that are rational expressions. Sums and differences are expressions to simplify. Equations are solved. Uses of the LCD When adding or subtracting rational expressions, keep the LCD throughout the simplification. When solving an equation, multiply each side by the LCD so the denominators are eliminated. Slide 6.6-4

  4. CLASSROOM EXAMPLE 1 Distinguishing between Expressions and Equations Identify each of the following as an expression or an equation. Then simplify the expression or solve the equation. expression Solution: equation Slide 6.6-5

  5. Objective 2 Solve equations with rational expressions. Slide 6.6-6

  6. Solve equations with rational expressions. When an equation involves fractions, we use the multiplication property of equality to clear the fractions. Choose as multiplier the LCD of all denominators in the fractions of the equation. Recall from Section 6.1 that the denominator of a rational expression cannot equal 0, since division by 0 is undefined. Therefore, when solving an equation with rational expressions that have variables in the denominator, the solution cannot be a number that makes the denominator equal 0. Slide 6.6-7

  7. CLASSROOM EXAMPLE 2 Solving an Equation with Rational Expressions Solve, and check the solution. Check: Solution: The use of the LCD here is different from its use in Section 6.5. Here, we use the multiplication property of equality to multiply each side of an equation by the LCD. Earlier, we used the fundamental property to multiply a fraction by another fraction that had the LCD as both its numerator and denominator. Slide 6.6-8

  8. Solve equations with rational expressions. (cont’d) While it is always a good idea to check solutions to guard against arithmetic and algebraic errors, it is essential to check proposed solutions when variables appear in denominators in the original equation. Solving an Equation with Rational Expressions Step 1:Multiply each side of the equation by the LCDto clear the equation of fractions. Be sure to distribute to every term on both sides. Step 2:Solvethe resulting equation. Step 3:Checkeach proposed solution by substituting it into the original equation. Reject any that cause a denominator to equal 0. Slide 6.6-9

  9. CLASSROOM EXAMPLE 3 Solving an Equation with Rational Expressions Solve, and check the proposed solution. Solution: When the equation is solved, − 1 is a proposed solution. However, since x = − 1leads to a 0denominator in the original equation, the solution set is Ø. Slide 6.6-10

  10. CLASSROOM EXAMPLE 4 Solving an Equation with Rational Expressions Solve, and check the proposed solution. Solution: The solution set is {4}. Slide 6.6-11

  11. CLASSROOM EXAMPLE 5 Solving an Equation with Rational Expressions Solve, and check the proposed solution. Solution: Since 0 does not make any denominators equal 0, the solution set is {0}. Slide 6.6-12

  12. CLASSROOM EXAMPLE 6 Solving an Equation with Rational Expressions Solve, and check the proposed solution (s). Solution: or The solution set is {−4, −1}. Slide 6.6-13

  13. CLASSROOM EXAMPLE 7 Solving an Equation with Rational Expressions Solve, and check the proposed solution. Solution: The solution set is {60}. Slide 6.6-14

  14. Objective 3 Solve a formula for a specified variable. Slide 6.6-15

  15. CLASSROOM EXAMPLE 8 Solving for a Specified Variable Solve each formula for the specified variable. Solution: Remember to treat the variable for which you are solving as if it were the only variable, and all others as if they were contants. Slide 6.6-16

  16. CLASSROOM EXAMPLE 9 Solving for a Specified Variable Solve the following formula for z. Solution: When solving an equation for a specified variable, be sure that the specified variable appears alone on only one side of the equals symbol in the final equation. Slide 6.6-17

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