# 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

■Solution:

**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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