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rational equation extraneous solution rational inequality

Objective. Solve rational equations and inequalities. Vocabulary. rational equation extraneous solution rational inequality.

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rational equation extraneous solution rational inequality

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  1. Objective Solve rational equations and inequalities. Vocabulary rational equation extraneous solution rational inequality

  2. 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.

  3. 18 x Example 1: Solving Rational Equations Solve the equation x – = 3.

  4. 6 – (–3) – 18 18 18 18 (–3) 6 x x Example 1 Continued Check x – = 3 x – = 3 3 3 6 – 3 3 –3 + 6 3 3 3  3 3 

  5. Solve the equation = + 2. 10 4 3 x Check It Out! Example 1a

  6. Solve the equation + = – . 5 6 7 4 4 x Check It Out! Example 1b

  7. 6 x Check It Out! Example 1c Solve the equation x = – 1.

  8. 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.

  9. = 3x + 4 5x x – 2 x – 2 Example 2A: Extraneous Solutions Solve each equation.

  10. x 2 2x – 5 11 x – 8 x – 8 Example 2B: Extraneous Solutions Solve each equation. + =

  11. 16 2 x2 – 16 x – 4 Check It Out! Example 2a Solve the equation . =

  12. Solve the equation . = + x 6 1 x x – 1 x – 1 Check It Out! Example 2b

  13. A rational inequality is an inequality that contains one or more rational expressions. You can 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.

  14. Solve ≤ 3 x x–6 Example 5: Using Graphs and Tables to Solve Rational Equations and Inequalities

  15. Solve ≤ 4 x x–3 Check It Out! Example 5a

  16. Solve = –2 8 x+1 Check It Out! Example 5b

  17. Solve ≤ 3 algebraically. 6 x–8 Example 6: Solving Rational Inequalities Algebraically

  18. Solve ≥ –4 algebraically. 6 x–2 Check It Out! Example 6a

  19. Solve < 6 algebraically. 9 x+ 3 Check It Out! Example 6b

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