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Compound Inequalities

Logic Operators: OR & ANDInterval Notation

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There are two different notations, when you writeyour answers you can use inequality notation or

interval notation.

0 4 6 82-7-6-5-4-3-2 -1 1 5 73 2-7-6-5-4-3-2 -1 1 5 730 4 6 8[

1x

Inequality notationfor graphs shownabove.

[ 1, )x

Interval notationfor graphs shownabove.

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),1[

Let's have a look at the interval notation.

For interval notation you list the smallestx

canbe, a comma, and then the largestxcan be sosolutions are anything that falls between thesmallest and largest.

This means x is

unboundedabove

2-7-6-5-4-3-2 -1 1 5 730 4 6 8[

unbounded

The bracket before the 1 is square because this is greater than "or equal to" (solutioncan equal the endpoint).

The bracket after the infinity sign is rounded because the interval goes on forever(unbounded) and since infinity is not a number, it doesn't equal the endpoint (there isno endpoint).

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]4,2(

Let's try another one.

2-7-6-5-4-3-2 -1 1 5 730 4 6 8

Rounded bracket means

cannot equal -2

Squared bracket meanscan equal 4

The brackets used in the interval notation above are the same onesused when you graph this.

( ]

This means everything between 2 and 4 but not including -2

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2-7-6-5-4-3-2 -1 1 5 730 4 6 8

)4,(

Let's look at another one

This means x isunbounded

below

Notice how the bracket notation for graphing corresponds to the brackets ininterval notation.

Remember that square is "or equal to" and round is up to but not equal. Bythe infinity sign it will always be round because it can't equal infinity (that is

not a number).

This means the

largest x can be is 4but can't equal 4

)

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OR" compound inequality

3or2 < xx

There are two intervals to list when you list in

interval notation.

2-7-6-5-4-3-2 -1 1 5 730 4 6 8) [

)2,( ),3[

When the solution consists of more than one interval,we join them with a union sign.

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EXAMPLE Solve an or compound inequality

Solve 3x+ 5 11 or 5x 7 23 . Then graph the solution.

SOLUTION

A solution of this compound inequality is a solution ofeitherof its parts.

First Inequality Second Inequality

3x + 5 11

3x 6

x 2

Write firstinequality.

Subtract 5 from eachside.Divide each side by3.

5x 7 23

5x 30

x 6

Write secondinequality.

Add 7 to eachside.Divide each side by5.

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EXAMPLE Solve an or compound inequality

ANSWER

The graph is shown below. The solutions are all realnumbers less than or equal to 2orgreater than or equalto 6.

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EXAMPLE Solve an and compound inequality

Solve 4 < 6x 10

14.

Then graph the solution.

4 < 6x 10 14

4 + 10< 6x 10 + 10 14 + 10

6 < 6x 24

1

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EXAMPLE Write and use a compound inequality

Biology

A monitor lizard has atemperature thatranges from 18Cto34C. Write the rangeof temperatures as acompound inequality.Then write aninequality giving thetemperature range in

degrees Fahrenheit.

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EXAMPLE Write and use a compound inequality

SOLUTION

The range of temperatures Ccan be represented by the inequality 18 C34. LetFrepresent the temperature in degrees Fahrenheit.

18 C 34 Writeinequality.

18 3459

(F 32)

32.4 F 32 61.2

64.4 F 93.2

Substitute forC.9

5(F 32)

Multiply each expression by ,

the reciprocal of .

9

55

9

Add 32 to eachexpression.

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EXAMPLE Write and use a compound inequality

ANSWER

The temperature of the monitor lizard ranges from64.4Fto 93.2F.

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PRACTICE

Solve the inequality. Then graph the solution.

9. 1 < 2x + 7 < 19

ANSWER

The solutions are all real numbers greater than 4and less than 6.

4

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PRACTICE

Solve the inequality. Then graph the solution.

10.8 x 5 6

The solutions are all real numbers greater than andequal to 11 and less than and equal to 3.

ANSWER

11 x 3

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PRACTICE

Solve the inequality. Then graph the solution.

11.x + 4 9 orx 3 7

ANSWER

The graph is shown below. The solutions are all realnumbers.

less than or equal to 5orgreater than or equal to 10.

x 5 orx10

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PRACTICE

Solve the inequality. Then graph the solution.

12. 3x 1< 1 or2x + 5 11

x < 0 orx3

less than 0orgreater than or equal to 3.

ANSWER

The graph is shown below. The solutions are all realnumbers.

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PRACTICE

1

3.

WHAT IF? In Biology example, write a compound

inequality for a lizard whose temperature ranges from15Cto 30C. Then write an inequality giving thetemperature range in degrees Fahrenheit.

15 C 30 or59 F 86

ANSWER