1. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §8.5 Rational InEqualities

2. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §8.5 → PolyNomial InEqualities  Any QUESTIONS About HomeWork §8.5 → HW-43 8.5 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 3 Bruce Mayer, PE Chabot College Mathematics Rational InEqualities  Inequalities involving rational expressions are called rational inequalities.  Like polynomial inequalities, rational inequalities can be solved using test values.  Unlike polynomials, however, rational expressions often have values for which the expression is UNDEFINED.

4. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 4 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION: write the related equation by changing the ≥ symbol to =  Next solve the related equation:

5. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Solve  In the case of rational inequalities, we must always find any values that make the denominator 0. As noted previously this occurs when x = 3.  Now use 3 and 7 to divide the number line into intervals: 3 7 I II III

6. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Solve I: Test 0, 0 is not a solution, so interval I is NOT part of the solution set. 10 is NOT a solution, so interval III is not part of the solution set. 4 is a solution, so interval II is part of the solution set. II: Test 4, II: Test 10,

7. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Solve  The solution set includes the interval II. The endpoint 7 is included because the inequality symbol is ≥ and 7 is a solution of the related equation.  The number 3 is not included because (x + 5)/(x − 3) is undefined for x = 3.  Thus the soln set of the inequality:

8. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 8 Bruce Mayer, PE Chabot College Mathematics To Solve a Rational InEquality 1.Change the inequality symbol to an equals sign and solve the related equation. 2.Find any replacements for which the rational expression is UNDEFINED. 3.Use the numbers found in step (1) and (2) to divide the number line into intervals. 4.Substitute a test value from each interval into the inequality. If the number is a solution, then the interval to which it belongs is part of the solution set

9. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 9 Bruce Mayer, PE Chabot College Mathematics To Solve a Rational InEquality 5.Select the interval(s) and any endpoints for which the inequality is satisfied and write set-builder notation or interval notation for the solution set. If the inequality symbol includes an “equals” then the solutions from step (1) are also included in the solution set. Those numbers found in step (2) should be EXCLUDED from the solution set, even if they are solutions from step (1)

10. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION: Analyze separately the Numerator & Denominator to find Brk-Pts Num = 0 and Den = 0 Solve

11. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Solve  Makes 3 intervals (−∞, 1), (1, 4), & (4, ∞) The Interval/Sign Graph 34 06 521–1 00 – – – – –+ + + + + + +– – –  The expression is positive in the interval (1, 4) and it is undefined for x = 1 and is 0 for x = 4.  The solution set is {x | 1 < x ≤ 4} or in interval notation (1, 4].

12. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Solve  From the Interval/Sign Graph 34 06 521–1 00 – – – – –+ + + + + + +– – –  The Solution on the Number Line 1 < x ≤ 4, or (1, 4] ]( 34 06 521–1

13. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION: Find the values that make the denominator equal to 0. x − 2 = 0 → x = 2  Next Solve the Related Equation Note that x = 2 is Excluded From Soln

14. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Solve  Plot Break Points on the Number Line 10-9-7-5-313579-10-8-4048-10-26-6102 RegionIIIIIIIV Test-Pt−7−7−5−503 Result−1/3 ≥ 00.14 ≥ 0−12 ≥ 0 63  0 True/FalseFalseTrueFalseTrue IIIIIIIV  Make Region/Truth Table

15. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Solve  Use Truth Table to Discern Solution Interval(−∞, −6)[−6, −4][−4, −2](2, ∞) Test-Pt−7−7−5−503 Result−1/3 ≥ 00.14 ≥ 0−12 ≥ 0 63  0 True/FalseFalseTrueFalseTrue  Thus the Solution 10-9-7-5-313579-10-8-4048-10-26-6102 [ ] (  Using Interval Notation: [−6, −4] U (2, ∞)

16. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 16 Bruce Mayer, PE Chabot College Mathematics Rational vs. PolyNom InEquals  Rational InEqualities are Similar to the PolyNomial version in that we find BREAK POINTS by analyzing a PolyNomial (the NUMERATOR) that is set to Zero  In the case of the Rational Version we obtain ADDITIONAL Break-Pts when the DEMONINATOR is Equal to Zero

17. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 17 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §8.5 Exercise Set 42, 52, 56, 62  Solve Another Rational InEquality

18. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 18 Bruce Mayer, PE Chabot College Mathematics All Done for Today Cliff Diving Ballistics “Splash” Speed for 100ft dive ≈ 55 mph!!!!

19. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 19 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

20. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 20 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

21. BMayer@ChabotCollege.edu MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 21 Bruce Mayer, PE Chabot College Mathematics