2.5 Inequalities BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 1. The expression x < a describes all real numbers less than a. Let a and b

1

2.5 Inequalities

BobsMathClass.Com Copyright © 2010 All Rights Reserved.

1. The expression x < a describes all real numbers less than a.

Let a and b be any real numbers:

Example: x < 4 describes all real numbers less than 4 (i.e. 1, 3, 2.5, -5, -94 etc.)

2. The expression x > b describes all real numbers greater than b.

Example: x > -7 describes all real numbers greater than -7 (i.e. -6, -4, 0, 2.1, 132 etc.)

In this section, we will be using the inequality symbols: <, > , and .

1. a < b means a is less than b2. a > b means a is greater than b

3. a b means a is less than or equal to b4. a b means a is greater than or equal b

3. The expression x a describes all real numbers less than or equal to a.

Example: x 4 describes all real numbers less than or equal to 4. (i.e. 1, 2.5, -9 , 4 etc.)

4. The expression x b describes all real numbers greater than or equal to b.

Example: x -7 describes all real numbers greater than or equal to -7. (i.e. -6, 13, -7 etc.)

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2.5 Inequalities


Sketching the graph of an inequality

Graphing an inequality using the number line

1. Draw a number line and place the real number from the inequality on the number line near the middle. You don’t have to necessarily have zero on the number line. 2. Darken the line beginning at the placement of the number going in the same direction as the inequality sign. Less than - shade to the left.

Greater than - shade to the right. 3. If the inequality is a < or > (with no equal sign), then place a parenthesis “(” or “)” at the placement of the number going in the same direction as the darkened number line. These parenthesis specify “not equal”.

4. If the inequality is a or , then place a bracket “[” or “]” at the placement of the number going in the same direction as the darkened number line. These brackets specify “equal to”.

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2.5 Inequalities


Example 1. Graph the inequality: x > 73

1. Draw a number line and place 73 near the middle.2. Since x is greater than 73, darken to the right.

3. Use parenthesis since >.

73

(

Your Turn Problem #1

Graph x 3

-3

]

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2.5 Inequalities


Expressing an inequality in interval notation

It is often convenient to express solutions using interval notation.

We will use the infinity symbol, , to indicate that there is no right end point.

We will use the symbol, , to indicate that there is no left end point. ( is called “negative infinity)

If x b or x b, then we use brackets to indicate that b is included in the solution set.If x < b or x > b, then we use parenthesis to indicate that b is not included in the solution set.

Since the infinity symbol indicates there is no endpoint, we will use parenthesis for infinity.

Partial list of interval notations, along with the sets of graphs they represent.

Set Notation Graph Interval Notation

{x x a}

{x x a}

{x x b}

{x x b}

a

(

a

[

b)

b]

a,

[a, )

,b

( ,b]

Note: the vertical line between the x’s is read as “such that”.

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2.5 Inequalities


Example 2. Express the following in interval notation.

a) x < 2

b) x –4

c) x > 5

d) x 1

Answers:

( ,2)

( , 4]

(5, )

[ 1, )

Your Turn Problem # 2

a) x < 8 b) x –2 c) x > –5 d) x 3

Express the following in interval notation.

Answers: a) ( ,8) b) ( , 2] c) ( 5, ) d) [3, )

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2.5 Inequalities


Solving Inequalities: To solve inequalities, we will first consider two properties.

Addition Property of Inequality

For all real numbers a, b, and c, a > b if and only if a + c > b + c.Multiplication Property for Inequalities

a) For all real numbers a, b, and c, with c > 0, a > b if and only if ac > bc.b) For all real numbers a, b, and c, with c < 0, a > b if and only if ac < bc.Note: One important point to notice from the multiplication property is that

whenever multiplying or dividing both sides by a negative number, the direction of the inequality sign must be changed!

Example: Take the following true inequality: 3 < 7

-12 > -28

(-4)3 > 7(-4)

The direction of the inequality symbol must be changed so that the statement remains true!

Multiply both sides by –4.

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2.5 Inequalities


Solve x 4 7 and write in sE examp t nole 3 tat o. i n. 4 4 1. Use the Addition Property of Inequality.

Add –4 to both sides.x 11

Write the answer properly using set notation: x x 11

(we read this as “the set of all x such that x is less than or equal to –11”)


Solve: x 2 8 and write in set notation.

x x 6 Answer:

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2.5 Inequalities


Solve 3x 12 and write in seExampl t notae t 4 n.. io

Write the answer properly using set notation: x x 4

1To solve for x, multiply both sides by ,

3or divide both sides by 3.

Since we multiplied both sides by a negative number, the direction of the inequality symbol had to change.

4

x 4

( 3x) 11

21

3 3

3x 12


Solve: 2x 18

Answer: x x 9

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2.5 Inequalities


Example 5. Solve the inequality, then graph the solution on a number line and write the answer in interval notation. 3x 4 19

Solution: Solve as you would with a linear equation. Remember that if you multiply or divide by a negative number, change the direction of the inequality symbol.

4 4

3x 15

x 5

Next graph the solution on a number line and then write the answer in interval notation.

3 3

-5(

( 5, )


Solve the inequality, then graph the solution on a number line and write the answer in interval notation.

2x 3 13

Answer:

8

[[8, )

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2.5 Inequalities


Example 6. Solve the inequality, then graph the solution on a number line and write the answer in interval notation.

2(x 3) 3(x 4)

Next graph the solution on a number line and write the answer in interval notation.

2x 6 3x 12

x 18

x 18

Solution: Solve as you would with a linear equation. Remember that if you multiply or divide by a negative number, change the direction of the inequality symbol.

1 1

18

]

( ,18]



2(4x 1) 5(2x 1)

Answer: (7

,2

7 2

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2.5 Inequalities


Example 7. Solve the inequality, then graph the solution on a number line and write the answer in interval notation. 3(2x 1) 2(3x 4) 3(x 1)

Solution: Solve as you would with a linear equation.

Next graph the solution on a number line and write the answer in interval notation.

6x 3 6x 8 3x 3

5 3x 3

8 3x8 8

x or x3 3

] 8,3

83



5(2x 3) 2(5x 1) 7(x 3)

Answer: ] 4,7

47

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2.5 Inequalities


We have seen that when solving equations that contain fractions, it was usually best to clear the fractions by multiplying both sides of the equation by the Least Common Denominator. This approach is also used when working with inequalities that involve fractions.Example 8. Solve the inequality and express the solution sets in interval notation. 1 4

x x 134 3

3x 16x 156

13x 156

(12) (12)1 4x x

(12)13

4 3

1. Multiply both sides by the LCD.

3 4

x 12 The solution set is - , 12

13 13

2. Solve the inequality.


3 1x x 2

5 3

Solve the inequality and express the solution set in interval notation.

15,

2

Answer:

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2.5 Inequalities



2x 1 3x 2 18 5 2


3x 2 2x 1 SolveExample 9.

9 3 1

Multiply both sides by the LCD.

3

(9) (9)(3x 2) (2x 1)1

9 3

(9)

8x

38

The solution set is - , .3

3x 2 6x 3 9

3x 1 9

3x 8 Distribute and solve the inequality.

3 3

Answer:1

,14

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2.5 Inequalities


Solve Exa 0.08x 0.09(2xmple 10. ) 1.30

Solution: 0.08x 0.09(2x) 1.30

0.08x 0.18x 1.30

0.26x 1.30

x 5

The solution set is 5,

0.26 0.26



0.07x 0.08(x 100) 38

Answer: (200, )

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2.5 Inequalities


Compound Statements

We use the words and and or in mathematics to form compound statements. Compound statements that use and are called conjunctions.A conjunction is true if all of its component parts are true.

1. 2+5 = 7 and -4 < -2 This is a true conjunction.2. 4>2 and -4<-6 This is a false conjunction.3. 5 > 3 and 0 < 9 This is a true conjunction.4. -2 + 3 = -1 and 3 + 4 = 7 This is a false conjunction.

Examples:

Compound statements that use or are called disjunctions.A disjunction is true if at least one of its component parts is true.

1. 0.13 > 0.12 or 0.24 > 0.43 2 12. or -5 + (-3) = -8

3 2

3 13. or 5 + 3 = 8

4 4

2 24. or 5 + (-3) = 8

5 5

Examples

This is a true disjunction


This is a false disjunction


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2.5 Inequalities


Graphing Compound Inequalities

Definition

The intersection of two sets A and B (written A B) is the set of all elements that are both in A and in B. Using set builder notation, we can write

We will first state two necessary definitions to graph compound inequalities

A B x x A nd Ba x

Definition

The union of two sets A and B (written A B) is the set of all elements that are in A or in B, or in both. Using set builder notation, we can write

Next Slide

A B x x A or x B

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2.5 Inequalities


Example 11. Graph the solution set for the compound inequality x > 3 and x 7 and express the solution in interval notation.

Let’s graph x > 3 and x 7 near the number line.

We want the intersection of the two inequalities, in other words, where the two lines overlap. (Shown in red)

The solution in interval notation will be (3, 7]

(

]

Solution: The key word to recognize is the word “and”. The word “and” implies that we are looking for the intersection of the two inequalities. Which numbers will satisfy both conditions?

3 7


Graph the solution set for the compound inequality x -2 and x < 5. Then express the solution in interval notation.

Answer:-2 5

Interval Notation:

[– 2, 5)[ )

( ]

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2.5 Inequalities


[

(

Example 12. Graph the solution set for the compound inequality x>2 and x7. Then express the solution in interval notation.

Solution:

2 7

Let’s graph x > 2 and x 7 near the number line.

The solution in interval notation will be 7, .

We want the intersection of the two inequalities where the two lines overlap. (Shown in red.)

[


Graph the solution set for the compound inequality x 0 and x >3. Then express the solution in interval notation.

Answer:

3

(Interval Notation: (3, ∞)

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2.5 Inequalities


[

)

Example 13. Graph the solution set for the compound inequality x<-3 and x 4. Then express the solution in interval notation.

Solution:

-3 4

Let’s graph x < -3 and x 4 near the number line.

Because of the word “and”, we want the intersection of the two solution sets, where the two purple lines overlap. In this problem, the lines do not overlap. There is no intersection, and therefore, no solutions.

The solution set is the empty set, written as .


Graph the solution set for the compound inequality x 0 and x <-3. Then express the solution in interval notation.

Answer:

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2.5 Inequalities


Example 14. Graph the solution set for the compound inequality x < 3 or x > 7 and

express the solution in interval notation.Solution: The key word to recognize is the word “or”. The word “or” implies that the solution is all real numbers that satisfy either (or both) inequalitiesWhen graphing the union of the two inequalities, graph each of the inequalities. The graph of the solution set will be the union of the two sets (whatever is graphed from each inequality).

3 7

( )

The solution in interval notation will be (–, 3) (7, )

We use the symbol “” when the answer has two or more sets in the solution.


Graph the solution set for the compound inequality x < -2 or x 5 and express the solution in interval notation.

Answer:

-2 5

[ )

(–, –2) [5, )

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2.5 Inequalities


Example 15. Graph the solution set for the compound inequality x < 2 or x 7 and

express the solution in interval notation.Solution: The key word to recognize is the word “or”. The word “or” implies that the solution is all real numbers that satisfy either (or both) inequalitiesWhen graphing the union of the two inequalities, graph each of the

inequalities. The graph of the solution will be whatever is graphed from each inequality.

After graphing each inequality, we recognize that the graph of x<2 will be contained in x 7. Then writing the number “2” on the number line would be unnecessary.

The solution in interval notation will be (–, 7]

2 7

])

]


Graph the solution set for the compound inequality x > 8 or x 12 and express the solution in interval notation.

Answer: (8 (8, )

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2.5 Inequalities


For the following compound inequalities, first solve the inequality as shown in the previous lesson. Then the solution can be graphed and expressed in interval notation. Example 16. Solve the compound inequality and graph the solution. Express the solution sets in interval notation.

2x 1 3 and 2x 1 1

Solution: First solve each inequality.

The graph will be the set of numbers than satisfy both inequalities.

Interval Notation: (0,2)

0 2

( )

and 2x 1 3 2x 1 1 1 1

2x 4x 2

2x 0 x 0

1 1

Your Turn Problem #16Solve the compound inequality and graph the solution. Express the solution sets in interval notation.3x 1 7 and 2x 3 4

Answer: ( ] 1 8,

2 3 1

2 83

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2.5 Inequalities


Conjunctions are sometimes written in compact form when one side of one of the inequalities matches a side from the other inequality.

a 1 and a 7 can be written asExample 1 a: 7.

Note: A disjunction can not be written in compact form.

4 < 2x < 6Then to get the x by itself, divide all three “sides” by 2.

The goal is to get the x by itself. Subtract 3 on the left hand side, middle section, and on the right hand side.

1 2x 3 9 Example 17. Solve the compound inequality using the compact form. Express the solution in interval notation.

Solution:

2 < x < 3Therefore the solution is all real numbers greater than –2 and less than 3.

1 2x 3 9 3 3 3

2 2 2

( )-2 3

Interval Notation: (-2,3)

Your Turn Problem #17Solve the compound inequality using the compact form. Express the solution in interval notation.5 3x 1 1 Answer: Interval Notation: (– 2,

0]( ]-2 0

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2.5 Inequalities


7 < x 6

Then to get rid of the negative sign, divide all three sides by –1.

Subtract 2 on the left hand side, middle section, and on the right hand side.

Example 18. Solve the compound inequality using the compact form. Express the solution in interval notation. 5 2 x 8

Solution: 2 225 2 x 8

Therefore the solution is all real numbers greater than or equal to –6 and less than 7. (Don’t forget to change the direction of the inequality sign whenever multiplying or dividing by a negative number.)

7 > x 6

1 1 1

[ )-6 7

Interval Notation: [-6,7)


Solve the compound inequality using the compact form. Express the solution in interval notation.7 1 2x 5

[ ]-2 4

Interval Notation: [– 2, 4]

Answer:

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2.5 Inequalities


Let x represent the score Kevin needs on the fifth exam.Solution:

We will use the formula for average which is to add all scores and divide by the number of scores. Since the problem says 80 or better, we will use the inequality symbol to show greater than or equal to 80.

84+64+76+92+x 80

5

316 x80

5

316 x 400

x 84Kevin must receive a score of 84 of better.

Kevin had scores of 84, 64, 76, and 92 on his first four exams of the semester. What score must he obtain on the fifth exam to have an average of 80 or better for the five exams?

Example 19.

Application problems from previous sections typically had a solution of consisting of one or two numerical answers. But some problems request a range of numbers, like “More than 3 years.”, “Between 80º and 90º.”, etc.). Inequalities (instead of equations) will be used in these kinds of application problems.

(Multiply by 5 on both sides to clear fractions.)

Michael has scores of 52, 84, 65 and 74 on his first four math exams. What score must he make on the fifth exam to have an average of 70 or better for the five exams?


Answer: Michael must score at least 75 on the fifth exam.

The End.B.R.12-15-06

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2.5 Inequalities BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 1. The expression x < a describes all real numbers less than a. Let a and b