Hawkes Learning Systems: College Algebra

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Hawkes Learning Systems:

College Algebra

Section 2.2: Solving Linear Inequalities


Objectives

o Solving linear inequalities.

o Solving compound linear inequalities.

o Solving absolute values inequalities.






Solving Linear Inequalities

o If the equality symbol in a linear equation is replaced

   linear inequality .

o For example, a linear inequality in is of the form ax b c x o The solution of a linear inequality typically consists of some interval of real numbers described in set notation, graphically or with interval notation.




Solving Linear Inequalities

Cancellation Properties for Inequalities are true for all inequalities.

Property

A



B



A C B C

If C



0, A



B



A



C



B



C

If C



0, A



B



A



C



B



C

Description

Adding the same quantity to both sides of an inequality results in an equivalent inequality.

If both sides of an inequality are multiplied by a positive quantity , the sign of the inequality is unchanged .

If both sides of an inequality are multiplied by a negative quantity , the sign of the inequality is reversed .




Example 1: Linear Inequalities

Solve the following linear inequality.

Step 1: Distribute.

Step 2: Combine like terms.

 x



2)

   x )

 x

    x

 

18

   x the reversal of the inequality sign.



5 x

 

20 x



4

  




Example 2: Linear Inequalities

Solve the following linear inequality.

2 4





7



7

 v

14( v

2

 v

2



5



5

5)















9

9

4 v v

9 v

4





4

14 v



70



9 v

5 v



70 v



14

Solution is (



,14)


Graphing a Solution



o The solutions in the previous examples were described using interval notation, but graphing can also be used to describe solutions. o Like in interval notation, parentheses are used when endpoints are not included in the interval and brackets are used when the endpoints are included in the interval.

o For example,



4,

  is graphed as follows:

 

,14

 is graphed as follows:



4



14




Solving Compound Linear Inequalities

A compound inequality is a statement containing two inequality symbols, and can be interpreted as two distinct inequalities joined by the word “and”.

For example, in a course where the grade depends solely on the grades of 5 exams, the following compound inequality could be used to determine the final exam grade needed to score a B in the course.

80



92 65 71 80

 x



90

5




Example 3: Solving Compound Inequalities

Solve the compound inequality from the previous slide.

80



92 65 71 80

5

 x



90

Step 1: Multiply all 400



308 x 450

92 142 from all sides.

 

Note: If this compound inequality relates to test scores, as

 




Example 4: Solving Compound Inequalities

Solve the compound inequality.

12

  x

 

15

    x



15

Note: each inequality is reversed since we are dividing by a negative number!

  

5 x



25

2 x 5

5

2 x

5

Solution is







5,

2

5




Solving Absolute Value Inequalities

An absolute value inequality is an inequality in which some variable expression appears inside absolute value symbols. x zero on the real number line. This means that absolute value inequalities can be written without absolute x

 a

    a and x

 a

 x

  a or x

 a




Example 5: Solving Absolute Value Inequalities

Solve the following absolute value inequality.

5 y 3 2 9

5 y 3 7

 

5 y

 

7 Step 2: Rewrite the inequality without absolute values.

Step 3: Solve as compound inequality.

 

5 y



10



4





2

5



5

4

Solution is , 2









Solve the following absolute value inequality and graph the solution.

 x



8

 x

 

8



2 x

 

14 x



7 or or

6



2 x



8



2 x



2 x

 

1





1 or

Solution is



, 1

 

7,

 



7





Solve the following absolute value inequality

10 x 7 4 x 7 6

Solution is



The solution set is the empty set, as it is impossible for the absolute value of any expression to be negative.




Interlude: Translating Inequality Phrases

o Many real-world applications leading to inequalities involve notions such as “is not greater than”, “at least as great as”, “does not exceed”, and so on. o Phrases such as these all have precise mathematical translations that use one of the four inequality

   




Interlude: Translating Inequality Phrases

Sign Phrase

“is greater than” 







Example

The average temperature in New

Mexico during the month of May is greater than 75. x



75

“at least as great as” The average price for a large screen digital TV is at least $9000. x



9000

“is less than” The number of skiers on the mountain is less than 100. x



100

“is not greater than” In a quality test, the number of defective products cannot be greater than 5. x



5




Example 8: Translating Inequality Phrases

The average daily high temperature in Phoenix, Arizona over the course of three days exceeded 80 degrees. Given that the high on the first day was 78 and the high on the third day was 85, what was the minimum high temperature on the second day?

78 85



80

3

163 240 x



77

The high temperature on the second day was greater than

77.




Example 9: Translating Inequality Phrases

As a test for quality at a plant manufacturing silicon wafers, a random sample of 10 batches of 1000 wafers each must not detect more than 5 defective wafers per batch on average. In the first 9 batches tested, the average number of defective wafers per batch is found to be 4.89. What is the maximum number of defective wafers that can be found in the 10 th batch for the plant to pass the quality test?

   x

5

10





44.01

50 x



5.99

Hawkes Learning Systems: College Algebra

Hawkes Learning Systems:

College Algebra

Objectives

Solving Linear Inequalities

Solving Linear Inequalities

Example 1: Linear Inequalities

Example 2: Linear Inequalities

Graphing a Solution

Solving Compound Linear Inequalities

Example 3: Solving Compound Inequalities

Example 4: Solving Compound Inequalities

Solving Absolute Value Inequalities

Interlude: Translating Inequality Phrases

Interlude: Translating Inequality Phrases

Example 8: Translating Inequality Phrases

Example 9: Translating Inequality Phrases

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Hawkes Learning Systems: College Algebra