[email protected] mth55_lec-55_sec_8-5b_rational_inequal.ppt 1 bruce mayer, pe chabot college...
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[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§8.5 Rational§8.5 RationalInEqualitiesInEqualities
[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §8.5 → PolyNomial InEqualities
Any QUESTIONS About HomeWork• §8.5 → HW-43
8.5 MTH 55
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Bruce Mayer, PE Chabot College Mathematics
Rational InEqualitiesRational 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.
[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt4
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
SOLUTION: write the related equation by changing the ≥ symbol to =
53.
3
x
x
Note that 3.x
Next solve the related equation:
5( 3) ( 3) 3
3
xx x
x
5 3 9x x
7.x
[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt5
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve 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
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Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
I: Test 0,0 5 5 5
30 3 3 3
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.
4 5 99 3
4 3 1
II: Test 4,
10 5 153
10 3 7
II: Test 10,
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Bruce Mayer, PE Chabot College Mathematics
Example Example Solve 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:
[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt8
Bruce Mayer, PE Chabot College Mathematics
To Solve a Rational InEqualityTo 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
[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt9
Bruce Mayer, PE Chabot College Mathematics
To Solve a Rational InEqualityTo 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)
[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt10
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
SOLUTION: Analyze separately the Numerator & Denominator to find Brk-Pts
3
x 11
3
x 1 1 0
3 x 1 x 1
0
4 xx 1
0
4 x 0
x 4
x 1 0
x 1
Num = 0 and Den = 0Solve
[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt11
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
Makes 3 intervals (−∞, 1), (1, 4), & (4, ∞)• The Interval/Sign Graph
3 40 6521–1
0 0 – – – – – + + + + + + + – – – – – –
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].
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Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
From the Interval/Sign Graph
3 40 6521–1
0 0 – – – – – + + + + + + + – – – – – –
The Solution on the Number Line
1 < x ≤ 4, or (1, 4]
](3 40 6521–1
[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt13
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
SOLUTION: Find the values that make the denominator equal to 0.
x − 2 = 0 → x = 2 Next Solve
the RelatedEquation• Note thatx = 2 isExcludedFrom Soln
( 4)( 6)0
2
x x
x
( 4)( 6)
( 2) 0( 2)2
x xx x
x
( 4)( 6) 0x x
4 0 or 6 0x x 4 or 6x x
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Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
Plot Break Points on the Number Line
10-9 -7 -5 -3 -1 1 3 5 7 9-10 -8 -4 0 4 8-10 -2 6-6 102
Region I II III IV
Test-Pt −7 −5 0 3
Result −1/3 ≥ 0 0.14 ≥ 0 −12 ≥ 0 63 0
True/False False True False True
I II III IV
Make Region/Truth Table
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Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
Use Truth Table to Discern SolutionInterval (−∞, −6) [−6, −4] [−4, −2] (2, ∞)
Test-Pt −7 −5 0 3
Result −1/3 ≥ 0 0.14 ≥ 0 −12 ≥ 0 63 0
True/False False True False True Thus the Solution
10-9 -7 -5 -3 -1 1 3 5 7 9-10 -8 -4 0 4 8-10 -2 6-6 102[ ] (
Using Interval Notation: [−6, −4] U (2, ∞)
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Bruce Mayer, PE Chabot College Mathematics
Rational vs. PolyNom InEqualsRational 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
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §8.5 Exercise Set• 42, 52, 56, 62
SolveAnotherRationalInEquality
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Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
CliffDiving
Ballistics
h
ft
hrmiles
2
9429.
“Splash” Speed for 100ft dive ≈ 55 mph!!!!
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt20
Bruce Mayer, PE Chabot College Mathematics
Graph Graph yy = | = |xx||
Make T-tablex y = |x |
-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6
x
y
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt21
Bruce Mayer, PE Chabot College Mathematics
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls -5
-4
-3
-2
-1
0
1
2
3
4
5
-10 -8 -6 -4 -2 0 2 4 6 8 10
M55_§JBerland_Graphs_0806.xls
x
y