Solve compound inequalitiesSolve compound inequalities

Section 6.4Section 6.4

#43 The continuum is that which is divisible into

indivisibles that are infinitely divisible. Physics. Aristotle

ConceptConcept

The first three sections of this chapter dealt primarily The first three sections of this chapter dealt primarily with inequalities that are single sidedwith inequalities that are single sided

Today we’re going to talk about compound, or double Today we’re going to talk about compound, or double sided, inequalitiessided, inequalities

DefinitionsDefinitionsDon’t write this down…just reviewDon’t write this down…just review

SymbolsSymbols Greater thanGreater than Less thanLess than Greater than or equal toGreater than or equal to Less than or equal toLess than or equal to

Does not include the number

Include the number

Open Circle

Closed Circle

108 x 426 x

Solving InequalitiesSolving Inequalities Sometimes we have to realize that our inequalities are Sometimes we have to realize that our inequalities are

boundedbounded For instance, if we were writing an equation for today’s For instance, if we were writing an equation for today’s

temperature, we’d say that it has to be between 40 degrees temperature, we’d say that it has to be between 40 degrees and 26 degreesand 26 degrees

Or we could say that it is less than or equal to 40 and greater Or we could say that it is less than or equal to 40 and greater than or equal to 26than or equal to 26

We would express this compound inequality this wayWe would express this compound inequality this way

40T

26 40T

26T

Solving InequalitiesSolving Inequalities As well, we can express an inequality that is bounded As well, we can express an inequality that is bounded

the other way.the other way. For instance, if we were going to talk about the temperatures For instance, if we were going to talk about the temperatures

that we don’t want a substance to be under, we could say that that we don’t want a substance to be under, we could say that it needs to be either below 2 degrees C or above 50 degrees Cit needs to be either below 2 degrees C or above 50 degrees C

We would express this compound inequality this wayWe would express this compound inequality this way

50T

2 or 50T T

2T

Solving InequalitiesSolving Inequalities There for we can explain that AND statements can be There for we can explain that AND statements can be

expressed as eitherexpressed as either

and OR statements can be expressed as and OR statements can be expressed as

a x b

or x a x b

and x a x b or

4 10

6

x

x

Solving InequalitiesSolving Inequalities To solve these we simply solve them twiceTo solve these we simply solve them twice

3 4 10x

3 4

1

x

x

1 6x

Because this is an AND

statement we can either

combine or leave separate

7 10

3

x

x

Solving InequalitiesSolving Inequalities To solve these we simply solve them twiceTo solve these we simply solve them twice

2 7 or 7 10x x

2 7

9

x

x

-9 or 3x x

Because it’s an OR statement,

we have to leave it an OR

statement

GraphingGraphing We often talk about the concept that AND statements We often talk about the concept that AND statements

convergeconverge

1 7x

1 or 3x x

Therefore OR statements divergeTherefore OR statements diverge

ExampleExample Solve for xSolve for x

12 4 2 18x

ExampleExample Solve for xSolve for x

5 3 8 or 5 35 10x x

ExamplesExamples

10 3 5 3 5 25x or x

ExamplesExamples

13 2 6 45x

ExamplesExamples

14 7 21x

Last ExampleLast ExampleYou estimate you can read at least 8 history text You estimate you can read at least 8 history text

pages per day. What are the possible numbers pages per day. What are the possible numbers of days it will take you to read at most 118 of days it will take you to read at most 118 pages?pages?

Most Important PointsMost Important Points Difference between an and statement and an or Difference between an and statement and an or

statementstatement How to solve compound inequalitiesHow to solve compound inequalities How to graph compound inequalitiesHow to graph compound inequalities

HomeworkHomework

6.46.4

1-8, 23-27, 47-50, 551-8, 23-27, 47-50, 55