Unit 1A

Precalculus

Lesson 7: Absolute Value and Rational Inequalities

Have you ever owed someone money?

If you owed a friend «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»$«/mo»«mn»10«/mn»«mo».«/mo»«mn»00«/mn»«/mrow»«/mstyle»«/math», you likely would not think of that as minus «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»$«/mo»«mn»10«/mn»«mo».«/mo»«mn»00«/mn»«/mrow»«/mstyle»«/math»; that is, you would not think of it as a signed number. Instead, you might think of the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»$«/mo»«mn»10«/mn»«mo».«/mo»«mn»00«/mn»«/mrow»«/mstyle»«/math» you must earn to repay your friend. When only the size (or magnitude) of a number is to be considered, absolute values are often used. The absolute value of a real number is non-negative.

For instance, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced open=¨|¨ close=¨|¨»«mrow»«mo»§#8722;«/mo»«mn»10«/mn»«/mrow»«/mfenced»«mo»=«/mo»«mn»10«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi mathvariant=¨normal¨» «/mi»«mfenced open=¨|¨ close=¨|¨»«mn»7«/mn»«/mfenced»«mo»=«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced open=¨|¨ close=¨|¨»«mn»0«/mn»«/mfenced»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».

The absolute value of any real number is defined as follows.

If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math», then «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced open=¨|¨ close=¨|¨»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math», and if «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#60;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math», then «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced open=¨|¨ close=¨|¨»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math».