MTH 070 Elementary Algebra Chapter 2 Equations and Inequalities in One Variable with Applications...

MTH 070MTH 070Elementary AlgebraElementary Algebra

Chapter 2Chapter 2Equations and Inequalities in Equations and Inequalities in

One Variable with ApplicationsOne Variable with Applications

2.4 – Linear Inequalities2.4 – Linear Inequalities

Copyright © 2010 by Ron Wallace, all rights reserved.

InequalityInequality

A statement that one expressions A statement that one expressions is …is …

<, ≤, >, or ≥<, ≤, >, or ≥… … a second expression.a second expression.

2( 3) 5x Applications involving inequalities involve terms such as “at least” (≥) and “at most” (≤).

Solving an InequalitySolving an Inequality

Determine Determine ALLALL values of the values of the variable that makes the variable that makes the inequality a true statement.inequality a true statement.

2( 3) 5x Try some …

0? 1? -3? 10? 7? 5? 20? 8?

Solutions of InequalitiesSolutions of Inequalities Three possible forms …Three possible forms …

Simple inequalitySimple inequality x x 2 2 x < –3x < –3

Interval notationInterval notation [2, [2, ) ) (–(–, –3) , –3)

GraphGraph

2[

3)

Review – Solving EquationsReview – Solving Equations Eliminate grouping symbols.Eliminate grouping symbols. Combine like terms.Combine like terms. Eliminate termsEliminate terms

Add opposites to other sideAdd opposites to other side i.e. addition propertyi.e. addition property

Eliminate factorsEliminate factors Multiply by reciprocals on the other sideMultiply by reciprocals on the other side i.e. multiplication propertyi.e. multiplication property

What happens when you do these things to inequalities?

NO PROBLEM

!

The Addition PropertyThe Addition Propertyw/ Inequalitiesw/ Inequalities

a bWhat happens if you add the same amount of weight to both sides of the scale?


a cb c

The inequality relationshipremains the same.

Works with subtraction too!


If an expression is added to If an expression is added to (subtracted from) both sides (subtracted from) both sides of an inequality, the result of an inequality, the result will be an equivalent will be an equivalent inequality (i.e. same inequality (i.e. same solutions).solutions).

The Multiplication PropertyThe Multiplication Propertyw/ Inequalitiesw/ Inequalities

a bWhat happens if you multiply each weight by the same POSITIVE value?


acbc

The inequality relationshipremains the same.

Works with division too!

NOTE: C > 0


a bWhat happens if you multiply each weight by the same NEGATIVE value?


acbc

The inequality relationshipreverses its direction.

Works with division too!

NOTE: C < 0


If both sides of an inequality are If both sides of an inequality are multiplied or divided by a multiplied or divided by a positive positive expressionexpression, the result will be an , the result will be an equivalent inequality (i.e. same equivalent inequality (i.e. same solutions).solutions).

If both sides of an inequality are If both sides of an inequality are multiplied or divided by a multiplied or divided by a negative negative expressionexpression ANDAND the direction of the the direction of the inequality is reversedinequality is reversed, the result will be , the result will be an equivalent inequality (i.e. same an equivalent inequality (i.e. same solutions).solutions).

Switching SidesSwitching Sides

If a < b, then how is b related to If a < b, then how is b related to a?a?

b > ab > a

If a > b, then how is b related to If a > b, then how is b related to a?a?

b < ab < aLikewise for ≤ and ≥.

Solving Inequalities – StrategySolving Inequalities – Strategy(just like equations w/ one exception)(just like equations w/ one exception)

Eliminate grouping symbolsEliminate grouping symbols Combine like termsCombine like terms Addition principle for inequalitiesAddition principle for inequalities Multiplication principle for inequalitiesMultiplication principle for inequalities

Careful w/ this one!Careful w/ this one! If the variable is on the right; switch sidesIf the variable is on the right; switch sides

Don’t forget to reverse the inequality symbol.Don’t forget to reverse the inequality symbol.

Basic Goal: x > ? or x < ? or x ≥ ? or x ≤ ?

Checking SolutionsChecking Solutions Need to check 3 values … Need to check 3 values … (okay, maybe 2 (okay, maybe 2

will do)will do) Assume the solution: x < 3Assume the solution: x < 3

Check x = 3 … this should make both sides Check x = 3 … this should make both sides equalequal

Check any value less than 3 … this should Check any value less than 3 … this should make the original inequality TRUE.make the original inequality TRUE.

Check any value greater than 3 … this Check any value greater than 3 … this should make the original inequality FALSEshould make the original inequality FALSE

Either of these

will do.

Hint: When checking inequalities, always check the number 0.

Solving & Checking InequalitiesSolving & Checking InequalitiesExample 1 of 5

Give solutions in all three forms

5 125x



3 12x



2 7 3x



4 8 3x x



323 42 1x x

MTH 070 Elementary Algebra Chapter 2 Equations and Inequalities in One Variable with Applications...

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