Rational Equations and Inequalities

■Section 3.6

Rational Functions are useful models for
■Inverse variation relationships
■Proportion or ratio problems
■Relationships approaching limits

Model: Product Exchange

Gas Oil
0 100
10 37.69
20 22.86
30 16.21
40 12.43
50 10
60 8.30
70 7.05
80 6.09
90 5.32
100 4.71


■Product exchange functions give the relationship between quantities of two items that can be produced by the same factory. An oil refinery can produce gasoline, heating oil, or a combination of the two . Data was collected for gas versus heating oil production in 1000 gallon allotments
■Determine a model for the product-exchange of gas and heating oil.

■View the graph in the next slide and address these questions
■What functions have we studied that have the same concavity and decreasing trend as the data?
■What in the context of the problem indicates choosing one of these functions over the other?

Product Exchange

Rational Function Model
■Serves as good model for data that approaches a fixed value or a limiting value since rational functions have asymptotes.

■Serves as a good model when the context is a ratio or comparison , such as gas versus heating oil

■How do we determine a rational model?
■Derive 5and Graphing calculators do not fit rational functions to data
■Estimate model using basic rational power function y = A/x. Will A > 1 or A < 1?
■Solution: Using graphing technology and eyeballing the curve of best fit we get

■Translate to start at (0,100)

Power Function Models

Actual Model

■Difficult to find actual model
Linearize the data (using Exercise 51 or 53)
■Determine from other problem context not provided

Rational Equations
■Determine the heating oil production if the gas production is 15 of the 1,000 gallon allotments.

■How do we solve rational equations?

Solving Rational Equations

Convert the rational equation into a polynomial equation by multiplying by the least common denominator

■Solve the resulting polynomial equation

■Check the solutions to be sure none of them are extraneous

■Use graphic methods if analytic methods fail

■Solve the following example

■Class Participation Activity

Solving Inequalities
■Solve inequalities with higher degree polynomial, radical, absolute value, or rational expressions using the graphic method

■Either set to zero and write one related function

■Or write a related equation for each side of the inequality

Solving Inequalities –
Graphic Method

Transformation of
Algebraic Functions

■Given any algebraic function y = f(x) we can transform it.

■a > 1 stretch, 0 < a < 1 compress, a < 0 reflection in x-axis
■b > 0 translation left, b < 0 translation right
■c > 0 translation up, c < 0 translation down

■Try translating the following function

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