[email protected] MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 1 Bruce Mayer, PE Chabot College...

[email protected] • MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§8.5 Rational§8.5 RationalInEqualitiesInEqualities



Review §Review §

Any QUESTIONS About• §8.5 → PolyNomial InEqualities

Any QUESTIONS About HomeWork• §8.5 → HW-43

8.5 MTH 55



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.



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




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




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,




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:



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



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)




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




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].




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




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




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




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, ∞)



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



WhiteBoard WorkWhiteBoard Work

Problems From §8.5 Exercise Set• 42, 52, 56, 62

SolveAnotherRationalInEquality



All Done for TodayAll Done for Today

CliffDiving

Ballistics

h

ft

hrmiles

2

9429.

“Splash” Speed for 100ft dive ≈ 55 mph!!!!



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22



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



-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

Date post:	17-Jan-2016
Category:	Documents
Upload:	ashley-bridges
View:	217 times
Download:	0 times

[email protected] MTH55_Lec-55_sec_8-5b_Rational_InEqual.ppt 1 Bruce Mayer, PE Chabot College...

Documents