An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs:

Symbol Expression Words

< A < B A is less than B

> A > B A is greater than B

< A < B A is less than orequal to B

> A > B A is greater than orequal to B

A is not equal to B

When am I ever going to use it?

Your parents and grandparents want you to start eating a healthy breakfast. The table shows the nutritional requirements for a healthy breakfast cereal with milk.

Healthy Breakfast Cereals (per serving)

Fat Less than 3 gramsProtein More than 5 gramsFiber At least 3 gramsSugar At most 5 grams

1. Suppose your favorite cereal has 2 grams of fat, 7 grams of protein, 3 grams of fiber and 4 grams of sugar. Is it a healthy cereal?

Answ

er

2. Is a cereal with 3 grams of fiber considered healthy?

Answ

er



3. Is a cereal with 5 grams of sugar considered healthy?

Answ

er



Graphing and WritingInequalities

with One Variable


Graphing and WritingInequalities with One Variable

Objectives

Identify solutions of inequalities with one variable.

Write and graph inequalities with one variable.

When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.

Important Words

Sample Sentence

Equivalent Translation

is more than Trenton is more than 10 miles away.

d > 10

is greater than

A is greater than B.

A > B

must exceed The speed must exceed 25 mph.

The speed is greater than 25 mph.

s > 25

When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.

Important Words

Sample Sentence

Equivalent Translation

cannot exceed

Time cannot exceed 60 minutes.

Time must be less than or equal to 60 minutes.

t < 60

is at most At most, 7 students were late for class.

Seven or fewer students were late for class.

n < 7

is at least Bob is at least 14 years old.

Bob's age is greater than or equal to 14.

B > 14

How are these inequalities read?

2 + 2 > 3  Two plus two is greater than 3

2 + 2 ≥ 4  Two plus two is greater than or equal to 4

2 + 2 < 5  Two plus two is less than 5

2 + 2 ≤ 5 Two plus two is less than or equal to 5

2 + 2 ≤ 4  Two plus two is less than or equal to 4

2 + 2 > 3  Two plus two is greater than or equal to 3

Writing inequalities

Let's translate each statement into an inequality.

x is less than 10

20 is greater than or equal to y

x < 10

words

inequality statement

translate to

20 > y

You try a few:

1. 14 is greater than a

2. b is less than or equal to 8

3. 6 is less than the product of f and 20

4. The sum of t and 9 is greater than or equal to 36

5. 7 more than w is less than or equal to 10

6. 19 decreased by p is greater than or equal to 2

7. Fewer than 12 items

8. No more than 50 students

9. At least 275 people attended the play

Answ

ers

Do you speak math?

Try to change the following expressions from English into math.

Twice a number is at most six.

Two plus a number is at least four.

2x ≤ 6

2 + x ≥ 4

Answer

Answer

Three less than a number is less than three times that number.

The sum of two consecutive numbers is at least thirteen.

Three times a number plus one is at least ten.

x - 3 < 3x

x + (x + 1) ≥ 13

3x + 1 > 10

Answer

Answer

Answer

A solution to an inequality is NOT a single number. It will have more than one value.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

This would be read as the solution set is all numbers greater than or equal to negative 5.

Solution Sets

Let's name the numbers that are solutions of the given inequality.

r > 10 Which of the following are solutions? {5, 10, 15, 20}

5 > 10 is not trueSo, not a solution

10 > 10 is not trueSo, not a solution

15 > 10 is trueSo, 15 is a solution

20 > 10 is trueSo, 20 is a solution

Answer:{15, 20} are solutions of the inequality r > 10

Let's try another one.

30 ≥ 5d; {4,5,6,7,8}

30 ≥ 5d30 ≥ 5(4)30 ≥ 20

30 ≥ 5d30 ≥ 5(5)30 ≥ 25

30 ≥ 5d30 ≥ 5(6)30 ≥ 30

30 ≥ 5d30 ≥ 5(7)30 ≥ 35

30 ≥ 5d30 ≥ 5(8)30 ≥ 40

Answer: {4,5,6}

Graphing Inequalities with Greater/Less Than or Equal To

An open circle on a number shows that the number is not part of the solution.It is used with "greater than" and "less than".The word equal is not included.< >

A closed circle on a number shows that the number is part of the solution.It is used with "greater than or equal to" and "less than or equal to".< >

Remember!

Open circle means that number is not included in the solution set and is used to represent < or >.

Closed circle means the solution set includes that number and is used to represent ≤ or ≥.

Do you know where to start?

How do represent the starting point?

Is there a special symbol?

Graphing Inequalities

Step 1: Figure out what the inequality solution requires. For example, rewrite x is less than one as x < 1.

Step 2: Draw a circle on the number line where the number being graphed is represented. In this case, an open circle since it represents the starting point for the inequality solution but is not part of the solution.

-1 0-2-3-4-5 1 2 3 4 5


Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality.

Step 3: Draw an arrow on the number line showing all possible solutions. For numbers greater than the variable, shade to the right of the boundary point. For numbers less than the variable, shade to the left of the boundary point.

-1 0-2-3-4-5 1 2 3 4 5

-1 0-2-3-4-5 1 2 3 4 5

x < 1

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Step 1: Figure out what the inequality solution requires. For example, rewrite x is greater than or equal to one as x > 1.

Step 2: Draw a circle on the number line where the number being graphed is represented. In this case, a closed circle since it represents the starting point for the inequality solution and is a part of the solution.


10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

You try

Graph the inequalityx > 5

Graph the inequality -3 > x

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Try these.Graph the inequalities.

1. x > 4  

-1 0-2-3-4-5 1 2 3 4 5

2. x < -5

-1 0-2-3-4-5 1 2 3 4 5

Try these.State the inequality shown.

1.   

-1 0-2-3-4-5 1 2 3 4 5

-1 0-2-3-4-5 1 2 3 4 5

2.   

7 Would this solution set be x > 4?

True

False

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Remember!

Closed circle means the solution set includes that number and is used to represent ≤ or ≥.

Open circle means that number is not included in the solution set and is used to represent < or >.

-1 0-2-3-4-5 1 2 3 4 58

A x > 3

B x < 3

C x < 3

D x > 3

5 6 7 8 9 10 11 12 13 14 159

A 11 < x

B 11 > x

C 11 > x

D 11 < x

-1 0-2-3-4-5 1 2 3 4 510

A x > -1

B x < -1

C x < -1

D x > -1

-1-5 1 511

A -4 < x

0-2-3-4 2 3 4

B -4 > x

C -4 < x

D -4 > x

12

A x > 0

-1 0-2-3-4-5 1 2 3 4 5

B x < 0

C x < 0

D x > 0

0  1  2  3  4  5  6  7  8  9  10

7.5

$7.50

7.5

at least

>

An employee earns

e

A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions.

Let e represent an employee's wages.

Try this:

The speed limit on a road is 55 miles per hour. Define a variable, write an inequality and graph the solution.

Answ

er

13 The sign shown below is posted in front of a roller coaster ride at theWadsworth County Fairgrounds. If h represents the height of a rider in inches, what is a correct translation of the statement on this sign?

A h < 48

B h > 48

C h ≤ 48

D h ≥ 48

All riders MUST beat least 48 inches tall.

Simple Inequalities Involving Addition

and Subtraction


  Objectives

Solve one-step inequalities by using addition.

Solve one-step inequalities by using subtraction.

x + 3 = 13 - 3 - 3 x = 10

Who remembers how to solve an algebraic equation?

Does 10 + 3 = 13 13 = 13Be sure to check your

answer!

Use the inverse of addition

Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequalities and inverse operations.

12 > x + 6

To find the solution, isolate the variable x.

Remember, it is isolated when it appears by itself on one side of the equation.

Step 1: Since 6 is added to x and subtraction is the inverse of addition, subtract 6 from both sides to undo the addition.

12 > x + 6- 6 - 6

6 > x

Step 2: Check the computation. Substitute the end point of 6 for x. The end point is not included (open circle) since x < 6.

12 > x + 612 > 6 + 612 > 12

0  1  2  3  4  5  6  7  8  9  10

Step 3: Check the direction of the inequality. Choose a number from your line (such as 4) and check that it fits the inequality.

0  1  2  3  4  5  6  7  8  9  10

6 > x6 > 4

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

k > -5

- 3 -3k + 3 > -2

A. k + 3 > -2

Solve and graph.

-5 is not included in solution set; therefore we graph with an open circle.

r > 11

+ 9 +9

r - 9 > 2

B. r - 9 > 2

Solve and graph.

1110 12 13 149876543210

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

5 > w

- 4 - 49 > w + 4

w < 5

C. 9 > w + 4

Solve and graph.

-1 0-2-3-4-5 1 2 3 4 5

5 62

A

-1 0-2-3-4-5 1 2 3 4 5B

-1 0-2-3-4-5 1 2 3 4 5C

-1 0-2-3-4-5 1 2 3 4 5D

1 2

1 3

14 Solve the inequality and graph the solution.

n - 2 >

2

2

2

5 6

5 6

5 6

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10A

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10B

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

C

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

D


2 < s + 8

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10A

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10B

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10C

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

D


-6 + b < -4

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10A

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10B

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10C

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

D


-5 > b - 2

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1.5

1.5

A

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10B

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1.5

C

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

D


3.5 < m + 2

1.5

Simple Inequalities Involving Multiplicationand Division


  Objectives

Solve one-step inequalities by using multiplication.

Solve one-step inequalities by using division.

Since x is multiplied by 3, divide both sides by 3 for the inverse operation.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Multiplying or Dividing by a Positive Number

3x > -27

3x > -27 3 3

x > -9

Remove for Graph

Solve the inequality and graph the solution.

2 3 r < 6

3 2( )

r < 9

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Since r is multiplied by 2/3, multiply both sides by the reciprocal of 2/3.

2 3 r < 6 3

2( )

Remove for Graph

19 4k > 24

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

A

B

C

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10D

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

20

A

B

C

-50 > 5q

10 > q

-10 < q

-10 > q

D 10 < q

21 X 2

A

B

C

D

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

< -1

22

A

B

C

g > 27

g > 36

g > 108

g > 36

D g > 108

3 4

23

A

B

C

-28 > 4d

d > -7

d > -7

d < -7

D d < -7

Sometimes you must multiply or divide to isolate the variable.

Multiplying or dividing both sides of an inequality by a negative number gives a surprising result.

Now let's see what happens when we multiply or divide by negative numbers.

1. Write down two numbers and put the appropriate inequality (< or >) between them.

2. Apply each rule to your original two numbers from step 1 and simplify. Write the correct inequality(< or >) between the answers.

 A. Add 4

 B. Subtract 4

 C. Multiply by 4

 D. Multiply by -5

 E. Divide by 4

 F. Divide by -4

3. What happened with the inequality symbol in your results?

4. Compare your results with the rest of the class.

5. What pattern(s) do you notice in the inequalities?

How do different operations affect inequalities?

Write a rule for inequalities.

Let's see what happens when we multiply this inequality by -1.

5 > -1

-1 • 5 ? -1 • -1 -5 ? 1

-5 < 1

We know 5 is greater than -1

Multiply both sides by -1

Is -5 less than or greater than 1?

You know -5 is less than 1, so you should use <

What happened to the inequality symbol to keep the inequality statement true?

Words OriginalInequality

Multiply/Divide by aNegative #

Result

Multiplying or dividing by a negative number reverses the inequality symbol

3 > 1 Multiply by-2

-6 < -2

-4 < 12 Divide by -4 1 > -3

The direction of the inequality changes only if the number you are using to multiply or divide by is negative.

Helpful Hint

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Dividing each side by -3 changes the > to <.

-3y > 15

-3y < 15 -3 -3

y < -5

Solve and graph.

A.

Divide each side by 7

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

7m < 21

7m < 21 7 7

m < 3

Solve and graph.

B.

Divide each side by 5.

5m > -25

5m > -25 5 5

m > -5

Solve and graph.

C.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

D. -8y > 24

Solve and graph.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

E. 9f > 45

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

You multiplied by a negative.

-r 2 < 5

-2( )r > -10

Multiply both sides by the reciprocal of -1/2.

-r 2 > 5 -2( )

Why did the inequality change?

1. -7h < 49

Try these.Solve and graph each inequality.

2. 3x > -15

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

3. 7m < 21


4. > -2

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

a -2

24

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve and graph.

2y < -4

25

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve and graph.

x -1 < -4

26

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve and graph.

-5y ≤ -25

27

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve and graph.

n -2 > 3

An inequality stays the same when you:

1. Add, subtract, multiply or divide by the same positive number on both sides

2. Add or subtract the same negative number on both sides

An inequality changes direction when you:

1. Multiply or divide by the same negative number on both sides

Solving Two-Step and Multiple-Step Inequalities


  Objectives

Solve inequalities that contain more than one operation.

Solve inequalities with variable terms on BOTH sides.

Now we'll solve some more complicated equations and inequalities

Ones that have two-step solutions because they involve two operations

Solving equations is like solving a puzzle. Keep working through the steps until you get the variable you're looking for alone on one side of the equation.

3x - 10 = 14

3x - 10 ≤ 14

You can solve two step inequalities in the same way you solve equations.

You can add any positive or negative number to both sides of the inequality.

You can multiply or divide both sides of an equality by any positive number.

3x - 10 ≤ 14 + 10 +10

3x < 24 3 3

x < 8

is solved in the same way as

REMEMBER! If you multiply or divide by a negative number, reverse the direction of the inequality symbol!

-3x ≤ 24 -3 -3

x ≥ -8

1. Solve this two-step equation.

  5 - 5x = 0

5 + -5x = 0 -5 -5

  -5x = -5  -5 -5

x = 1

Step 2: Use multiplicative    inverse 

Step 1: Use additive inverse 

2. Solve this two-step inequality.

  26 < 3n + 1  -1  - 1

  25 < 3n  3 3   8 < n  

1 3

Step 2: Use multiplicative    inverse 

Step 1: Use additive inverse 

4p - 9 ≥ 23 + 9 +9  Add 9 to both sides

4p ≥ 32  Divide both sides by 4 4 4 (sign stays the same)

p ≥ 8

Graph the solution { p | p ≥ 8 }

Solve 4p - 9 ≥ 23

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10 Move to reveal graph


1. 6 - x > 3

2. -4c + 16 < 0 10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10


3. -3y - 21 < 0

4. 22 < -5x + 18x - 4 10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

28

A

B

C

Solve and graph the solution.18 < 4(x + 2)

2.5 < x

2.5 > x

2.5 < x

D 2.5 > x

29

A

B

C

Solve and graph the solution.16 - x > 7x

2 < x

2 > x

2 < x

D 2 > x

30

A

B

C

Solve and graph the solution.8 < 5x + 3

1 < x

1 > x

1 < x

D 1 > x

31

A

B

C

Solve and graph the solution.12 + 5x < 32

x > 4

x < 4

x < 4

D x > 4

32

A

B

C

Solve and graph the solution.36 > -3(x - 5)

-7 < x

-7 > x

-7 < x

D

-7 > x

33 Which graph represents the solution set for:

A

B

C

D

Question from ADP Algebra I End-of-Course Practice Test

0 1 2-1-2

0 1 2-1-2

0 1 2-1-2

0 1 2-1-2

1 2 5 2 3 6x <

Find all negative odd integers that satisfy the following inequality: –3x + 1 17

34 Which value of x is in the solution set of

A 8

B 9

C 12

D 16

x + 5 < 17

35 What is the solution of 3(2m − 1) ≤ 4m + 7?

A m ≤ 5

B m ≥ 5

C m ≤ 4

D m ≥ 4

36 In the set of positive integers, what is the solution set of the inequality2x - 3 < 5?

A {0,1,2,3}

B {1,2,3}

C {0,1,2,3,4}

D {1,2,3,4}

37 The inequality

A x < – 5 6

B x > – 5 6

C x < 6

D x > 6

x + 3 < 2x - 6

38 Given: A = {18, 6, −3, −12}Determine all elements of set A that are in the solution of the inequality 2x + 3 < −2x − 7. 3

A 18

B 6

C -3

D -12

Your town is having a fall carnival. Admission into the carnival is $3.00 and each game inside costs $0.25.

Write an inequality that represents the possible number of games that can be played if you have $10.00.

What is the maximum number of games that can be played?

Hint:Ten dollars is the maximum amount of money that you have to spend at the carnival. What inequality symbol would be used?

ANSWER

.25x + 3 ≤ 10

.25x + 3 ≤ 10 - 3 -3

.25x ≤ 7 .25 .25

x ≤ 28

The maximum number of games that can be played is 28.

You have $65.00 in birthday money and want to buy some CDs and a DVD. Suppose a DVD cost $15.00 and a CD cost $12.00.

Write an inequality to find out how many CDs you can buy along with one DVD. Solve the inequality.

Hint 1The cost of 1 DVD and the unknown number of CDs must be less or equal to $65.

Hint 2How much does 1 CD cost? How would you express an unknown number of CDs?

Pull down the shade to see the answer.

15 + 12x ≤ 65

15 + 12x ≤ 65 -15 -15

12x ≤ 50

12x ≤ 50 12 12

x ≤ 4.16

Can you buy 0.16 of a CD?You can buy 4 CDs and 1 DVD.

Matt was getting ready for school. He had less than $150 to buy school clothes. Matt bought 3 pairs of pants and spent $30 on snacks and other items.

How much could one pair of pants cost, if they were all the same price? Write an inequality.

What do you know? Pull tab if you need help.

3x + 30 < 150   - 30 - 30

  3x < 120 3 3 x < 40

Answer

Hin

t

Try These

1. You have $60 to spend on a concert. Tickets cost $18 each and parking is $8. Write an inequality to model the situation.

Answ

ers

2. If you borrow the $60 from your mom and pay her back at a rate of $7 per week, when will your debt be under $15?

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Try This.

To earn an A in math class, you must earn a total of at least 180 points on three tests. On the first two tests, your scores were 58 and 59. What is the minimum score you must get on the third test in order to earn an A?

Define a variable, write an inequality and graph the solutions.

Answ

er

Thelma and Laura start a lawn-mowing business and buy a lawnmower for$225. They plan to charge $15 to mow one lawn. What is the minimum number of lawns they need to mow if they wish to earn a profit of at least $750?

39 Roger is having a picnic for 78 guests. He plans to serve each guest at least one hot dog. If each package, p, contains eight hot dogs, which inequality could be used to determine how many packages of hot dogs Roger will need to buy?

A p ≥ 78

B 8p ≥ 78

C 8 + p ≥ 78

D 78 − p ≥ 8

40 A school group needs a banner to carry in a parade. The narrowest street the parade is marching down measures 36 ft across, but some space is taken up by parked cars. The students have decided the banner should be 18 ft long. There is 45 ft of trim available to sew around the border of the banner. What is the greatest possible width for the banner?

A w < 27

B w < 4.5

C w < 18

D w < 4.5

Solving CompoundInequalities


Objectives

Solve inequalities that contain more than one operation.

Graph solution sets of compound inequalities.


When two inequalities are combined into one statement by the words AND/OR, the result is called a compound inequality.

A solution of a compound inequality joined by and is any number that makes both inequalities true.

A solution of a compound inequality joined by or is any number that makes either inequality true.


Here are some samples

x > -2 AND x < 3 -2 < x < 3

x ≥ -2 AND x ≤ 3

-2 ≤ x ≤ 3-1 0-2-3-4 1 2 3 4

NOTE: "and" means intersection, so you graph the intersection of the two inequalities

-1 0-2-3-4 1 2 3 4

Here are some additional samples

x < -2 OR x > 3

x ≤ -2 OR x ≥ 3

NOTE: "or" means union, so you graph the union of the two inequalities


-1 0-2-3-4 1 2 3 4

-1 0-2-3-4 1 2 3 4

41 Which inequality is represented in the graph below?

A – 4 < x < 2

B – 4 x < 2

C – 4 < x 2

D – 4 x 2

–5 –4 –3 –2 –1 0 1 2 3 4 5

42 Which inequality is represented in the accompanying graph?

A –3 ≤ x < 4

B –3 ≤ x ≤ 4

C –3 < x < 4

D –3 < x ≤ 4

–3  0  4

4 ≤ x+2 ≤ 8 is the same as writing

4 ≤ x+2 AND x+2 ≤ 8

You will need to solve both of these inequalities and graph their intersection.

Solving Compound Inequalities that contain an AND statement

Let's solve it!

4 ≤ x+2 ≤ 8

4 ≤ x+2 AND x+2 ≤ 8

4 ≤ x+2  AND x+2 ≤ 8-2 -2  -2 -2 2 ≤ x AND x ≤ 6

2 < x < 6

Step 1 Rewrite as 2 separate inequalities

Step 2 Solve each inequality for x

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Step 3 Graph your solution

Let's try another one

-9 < x - 10 < 5

-9 < x-10 AND x-10 < 5

-9 < x-10 AND x-10 < 5+10 +10  +10 +10 1 < x AND x < 15

1 < x < 15

What do I do next?

And then what?

1  3   5   7   9   11  13   15

43 Which result below is correct for this inequality: -3 < x+2 < 7

A 1 < x < 5

B -5 < x < 5

C -3 > x > 5

Now let's look at the OR statements.

2 + r < 12 OR r + 5 > 19

Just like before, solve each one separately. However, with OR statements, graph their union.

2 + r < 12 OR r + 5 > 19  -2 -2  - 5 -5 r < 10 OR r > 14

r < 10 or r > 14

8  10   12   14   16   18  20   22

Pull

Compound Inequalities in Applied Problems

Let's start off by translating the words of an applied problem into math.

The sum of 3 times a number and two lies between 8 and 11.

"The sum of 3 times a number and two" translates into what?( Pull tab to see if you are correct...)

Here is another OR statement.

 7x ≥ 21 OR 2x ≤ -2

Solve each one separately, then graph their union.

 7x ≥ 21  OR 2x ≤ -2   7 7    2 2

  x ≥ 3   OR x ≤ -1

  x ≥ 3 or x ≤ -1

-3  -1   1   3   5   7   9   11

Writing a Compound Inequality From a Graph

How would you write this?

x ≤ -6 OR x ≥ 0Move to find out

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Writing a Compound Inequality From a Graph

How would you write this?

-5 < x < 2

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Move to find out

Try these.Solve and graph the solution set.

1. -18 < 3x - 6 < -3

2. -5x + 2 > 27 or x - 3 > 210 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Try these.Solve and graph the solution set.

3. -2x - 6 > 4 or x + 5 > 8

4. -6 < 2x + 4 < 1010 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

44 In order to be admitted for a certain ride at an amusement park, a child must be greater than or equal to 36 inches tall and less than 48 inches tall. Which graph represents these conditions?

A

B

C

D

24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

45 Which graph shows the solution to this compound inequality?

r - 1 < 0 or r - 1 > 4

A

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

B

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

C

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

46 Which graph represents the solution set for 2x - 4 ≤ 8 and x + 5 ≥ 7?

A

B

C

D

1 2 3 4 5 6 7

1 2 3 4

1 2 3 4

1 2 3 4

5 6 7

5 6 7

5 6 7

47 Solve -6 > -3x - 6 and -3x - 6 > 6

A 0 > x and x < -4

B 0 < x and x < -4

C 4 < x and x > -4

D 4 < x and x < -4

48 Solve 3x - 8 < 13 or -3x + 10 > 5

A x < 7 or x >

B x < or x >

C x < 7 or x <

5 3

5 3

D x < 7 or x >

5 3

5 3

5 3

49 The statement “x ≥ 4 and 2x - 4 < 6” is true when x is equal to

A 1

B 10

C 5

D 4

50 Write the inequality shown by the graph.

A x < -5 or x > 1

B x < -5 and x > 1

C 1 < x and x > -6

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

D x > -5 or x > 1

51 Write the inequality shown by the graph.

A x > -7 or x < 3

B x > -7 and x < 3

C x > -7 or x > 3

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

D x > -7 and x < 3

-100 -50  0  50  100  150  200  250 300 350 400

A cell phone plan offers free minutes for no more than 250 minutes per month. Define a variable and write an inequality for the possible number of free minutes. Graph the solution.

Let m = number of minutes

0 < m < 250

Abou

tTh

ink

The sum of 3 times a number and 2 lies between 8 and 11.

We found 3x + 2 but how will we translate "lies between 8 and 11"?

What inequality symbol will we use?

If 3x + 2 lies between 8 and 11, is it larger or smaller than 8?

Write an inequality. Pull tab to see if you are correct.

Pull

Solve the inequality.

8 < 3x + 2 < 11 - 2 - 2 - 2

6 < 3x < 9 3 3 3

2 < x < 3

The light rail train charges $2.00 a ticket. Children 6 and under ride for free. Children over 6 and under 12 pay half fare and senior citizens (people over 65) get 25% off.

Write an inequality to describe x, the ages in years of all those who are eligible to receive reduced fares.

Hin

t

Children who are 6 but less than 12 pay half. Children under 6 ride free. People 65 or older pay a reduced fare. How does that translate into an inequality?

Can we combine two groups? What inequality symbols will we use?Will this be an "and" inequality?Could it be an "or" inequality?

We can combine all the children under 12. We would use x < 12.

For people that are 65 or older we would use x ≥ 65.

x < 12 or x ≥ 65Does someone age 25 get a reduced fare?

Draw a graph to illustrate the inequality.

x < 12 or x ≥ 65

52 In 1999 a house sold for $145,000. The house sold again in 2009 for $211,000. Write a compound inequality that represents the different values that the house was worth between 1999 and 2008.

A 145,000 < H < 211,000

B 145,000 > H < 211,000

C 145,000 ≤ H ≤ 211,000

D 145,000 ≤ H ≥ 211,000

Special Cases of Compound Inequalities


Objectives

Recognize special cases of solution sets when solving compound inequalities.

Graph solution sets of no solution and Real Numbers.

Special Solutions

A solution of a compound inequality joined by and is any number that makes both inequalities true.

When there is no number that makes both inequalities true, we say there is no solution.

When all numbers make both inequalities true, we say the solution is the set of Real Numbers.

No Solution andthe Set of Real Numbers

2x > 18 AND -3x > 12 

2x > 18 AND -3x > 12 2 2 -3 -3

x > 9 AND x < -4

The solution set is No Solution since there are no numbers that are both > 9 and < -4.

We write this solution as { } or 0

Another Example

-2x + 3 > 17 OR 5(x + 2) > -40 

-2x + 3 > 17 OR 5x + 10 > -40 - 3 - 3   - 10 -10

-2x > 14 OR 5x > -50 -2 -2  5 5

x < -7  x > -10

The solution set is Reals since all numbers are either < -7 or > -10.We write this solution set as R.

-11   -10 -9 -8 -7 -6 -5 -4 -3

Try these.

1. 4(x + 3) < 8x - 12 and 2(x + 3) < x + 6  

2. -2(x - 2) < 10 or 5x + 7 < 3(5 + x)

3. 3x + 8 > 23 and -2(x - 2) > -14

4. 6x + 3 > 4x - 13 and 5x + 8 > 2(x + 19)

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