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8.5 Solving Rational Equations and Inequalities

Objective. Solve rational equations and inequalities. Vocabulary. rational equation extraneous solution rational inequality a. 8.5 Solving Rational Equations and Inequalities. x. =. 2. 3 x + 4. 2 x – 5. 11. 5 x. x – 2. x – 8. x – 2. x – 8.

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8.5 Solving Rational Equations and Inequalities

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  1. Objective Solve rational equations and inequalities. Vocabulary rational equation extraneous solution rational inequality a 8.5 Solving Rational Equations and Inequalities

  2. x = 2 3x + 4 2x – 5 11 5x x – 2 x – 8 x – 2 x – 8 Always check for extraneous solutions by substituting into the equation. Example 2A: Extraneous Solutions Solve each equation. + = Check:

  3. = + 2. 10 4 18 3 x x To eliminate the denominators, multiply each term of the equation by the least common denominator (LCD) of all of the expressions in the equation. Example 1: Solving Rational Equations x– = 3.

  4. Solve ≤ 3 by using a graph and a table. Use a graph. On a graphing calculator, Y1 = and Y2 = 3. x x–6 x x–6 Example 5: Using Graphs and Tables to Solve Rational Inequalities (9, 3) Which x-values gives us output y-values that are less 3? Vertical asymptote: x = 6

  5. 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.

  6. Solve ≤ 3 6 x–8 To solve rational inequalities algebraically multiply each term by the LCD of all the denominators. You must consider two cases: the LCD is positive or the LCD is negative. Example 6: Solving Rational Inequalities Algebraically Case 1 LCD is positive. Step 1 Solve for x. Step 2 Consider the sign of the LCD. For Case 1, the solution must satisfy x ≥ 10 and x > 8, which simplifies to x ≥ 10.

  7. Solve ≤ 3 6 x–8 6 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}. Example 6 Con’t Case 2 LCD is negative. Step 2 Consider the sign of the LCD. Step 1 Solve for x. For Case 2, the solution must satisfy x ≤ 10 and x < 8, which simplifies to x < 8.

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