8-5. A rational inequality is an inequality that contains one or more rational expressions. x x 6. 3 by using a graph and a table.
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1 A rational inequality is an inequality that contains one or more rational expressions. x x 3 by using a graph and a table. Use a graph. On a graphing calculator, Y1 = x and Y = 3. x The graph of Y1 is at or below the graph of Y when x < or when x 9. (9, 3) Vertical asymptote: x =
2 Example 5 Continued Use a table. The table shows that Y1 is undefined when x = and that Y1 Y when x 9. The solution of the inequality is (-, ) U [9, ).
3 x x 3 Check It Out! Example 5a 4 by using a graph and a table. Use a graph. On a graphing calculator, Y1 = x and Y = 4. x 3 The graph of Y1 is at or above the graph of Y when x > 3 and when x 4. (4, 4) Vertical asymptote: x = 3
4 Check It Out! Example 5a continued Use a table. The table shows that Y1 is undefined when x = 3 and that Y1 Y when x 4. The solution of the inequality is (-3, 4].
5 To solve rational inequalities algebraically, start by multiplying by the (LCD). You must consider two cases: the LCD is positive and the LCD is negative.
6 Example : Solving Rational Inequalities Algebraically x 8 3 algebraically. Case 1 LCD is positive. Step 1 for x. x 8 (x 8) 3(x 8) Multiply 3x x 10 x x 10 by the LCD. Simplify. Note that x 8. for x. Rewrite with the variable on the left.
7 Example Continued x 8 3 algebraically. Step Consider the sign of the LCD. x 8 > 0 x > 8 LCD is positive. for x. For Case 1, the solution must satisfy x 10 and x > 8, which simplifies to x 10.
8 Example : Solving Rational Inequalities Algebraically x 8 3 algebraically. Case LCD is negative. Step 1 for x. x 8 (x 8) 3(x 8) Multiply 3x x 10 x x 10 by the LCD. Reverse the inequality. Simplify. Note that x 8. for x. Rewrite with the variable on the left.
9 Example Continued x 8 Step Consider the sign of the LCD. x 8 < 0 x < 8 3 algebraically. LCD is negative. for x. For Case, the solution must satisfy x 10 and x < 8, which simplifies to x < 8. The solution set of the original inequality is the union of the solutions to both Case 1 and Case. The solution is (-, 8) U [10, ).
10 Check It Out! Example a x 4 algebraically. Case 1 LCD is positive. Step 1 for x. x (x ) 4(x ) Multiply 4x + 8 4x 1 x x 1 by the LCD. Simplify. Note that x. for x. Rewrite with the variable on the left.
11 Check It Out! Example a Continued x 4 algebraically. Step Consider the sign of the LCD. x > 0 LCD is positive. x > for x. For Case 1, the solution must satisfy and x >, which simplifies to x >. x 1
12 Check It Out! Example a Continued x 4 algebraically. Case LCD is negative. Step 1 for x. x (x ) 4(x ) Multiply by the LCD. Reverse the inequality. 4x + 8 Simplify. Note that x. 4x 1 x x 1 for x. Rewrite with the variable on the left.
13 Check It Out! Example a Continued 4 algebraically. x Step Consider the sign of the LCD. x < 0 LCD is negative. x < for x. For Case, the solution must satisfy and x <, which simplifies to x 1. x 1 The solution set of the original inequality is the union of the solutions to both Case 1 and Case. The solution is (-, ½] U (, ).
14 Check It Out! Example b 9 x + 3 > algebraically. Case 1 LCD is positive. Step 1 for x. 9 x + 3 (x + 3) > (x + 3) Multiply 9 > x > x 3 > x x < 3 by the LCD. Simplify. Note that x 3. for x. Rewrite with the variable on the left.
15 Check It Out! Example b Continued 9 x + 3 > algebraically. Step Consider the sign of the LCD. x + 3 > 0 LCD is positive. x > 3 for x. For Case 1, the solution must satisfy x < 3 and x > 3. Therefore, (-3, -3/) is the solution for Case 1.
16 Check It Out! Example b Continued 9 x + 3 Case LCD is negative. Step 1 for x. > algebraically. 9 Multiply by the LCD. x + 3 (x + 3) < (x + 3) Reverse the inequality. 9 < x + 18 Simplify. Note that x 3. 9 < x 3 < x x > 3 for x. Rewrite with the variable on the left.
17 Check It Out! Example b Continued 9 x + 3 Step Consider the sign of the LCD. x + 3 < 0 < algebraically. The solution set of the original inequality is the union of the solutions to both Case 1 and Case. The solution is (-3, -3/). LCD is negative. x < 3 for x. For Case, the solution must satisfy x > 3 and x < 3. Therefore, there are no solutions for Case.
18 HW pg. 05 # s 30-3, 58-1
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