Unit 1A

Precalculus

Lesson 8: Quadratic Inequalities

Quadratic inequalities can be used to solve many real-life problems. Imagine a farmer wishing to create a rectangular livestock pen with a minimum area of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»100«/mn»«mo»§#160;«/mo»«msup»«mi mathvariant=¨normal¨»m«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math». She plans to use «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»50«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math» of fence and wants to know the possible dimensions of the pen.

Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» be the length of the rectangular pen. There is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»50«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math» of fence, so one length and one width will use «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»25«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math». As such, the width can be expressed as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»25«/mn»«mo»-«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» and the area can be expressed as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»A«/mi»«mo»=«/mo»«mi»x«/mi»«mfenced»«mrow»«mn»25«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math».

The problem to be solved can be represented by the inequality «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mfenced»«mrow»«mn»25«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#8805;«/mo»«mn»100«/mn»«/mrow»«/mstyle»«/math».

This quadratic inequality can be expanded and simplified as follows.


There are three distinct ways to determine the solution to this inequality.

1. graphically
2. algebraically
3. using sign analysis

Each method can be used in any of the following examples, but some methods are easier to use in certain cases.