Table of Contents. Key Terms. Pre-Assessment. Graphing/Writing with One Variable. Simple Inequalities Addition/Subtraction. Simple Inequalities Multiplication/Division. Two-Step & Multiple-Step. Compound Inequalities. Special Cases of Compound Inequalities. Key Terms.
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Table of Contents
Key Terms
Pre-Assessment
Graphing/Writing with One Variable
Simple Inequalities Addition/Subtraction
Simple Inequalities Multiplication/Division
Two-Step & Multiple-Step
Compound Inequalities
Special Cases of Compound Inequalities
Key Terms
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Key Terms
Equation
Inequality
Solution Set
Compound Inequality
Set Builder Notation
Pre-Assessment
What do you already know?
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1
Evaluate the expression for a=2 and b=6.
b - a
2
Compare and order. Write <, > or =.
1 [ ] 2
3 5
A
<
B
>
C
=
3
Simplify the expression by combining like terms.
4x - 5 + 3x + 2
A
4x
B
12x + 2
C
7x - 3
D
7x + 7
4
Simplify the expression
(3 - d)5
A
3 - 5d
B
15 - 5d
C
15 + 5d
D
15 - d
5
Solve r - 8 = -13
6
There are 461 students and 20 teachers taking buses on a trip to a museum. Each bus can seat a maximum of 52. What is the least number of buses needed for the trip?
A
8
B
9
C
10
D
11
What do these symbols mean?
Less
Than
Greater
Than
Less Than
or Equal To
Greater Than
or Equal To
move square to reveal answer
An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs:
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.
Did you compare your favorite cereal's nutritional values to the healthy requirements on the table?
Answer
Healthy Breakfast Cereals (per serving)
If you did, you found out that you have been eating a healthy breakfast. Now you can prove it to your parents and grandparents.
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?
Healthy Breakfast Cereals (per serving)
A cereal with 3 grams of fiber just makes it at being healthy. It needed at least 3 grams.
Answer
2. Is a cereal with 3 grams of fiber considered healthy?
Healthy Breakfast Cereals (per serving)
Answer
A cereal with 5 grams of sugar is still considered healthy but is very close to being unhealthy.
3. Is a cereal with 5 grams of sugar considered healthy?
Graphing and Writing
Inequalities
with One Variable
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Graphing and Writing
Inequalities 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.
When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.
How are these inequalities read?
2 + 2 > 3 Two plus two is greater than 3
2 + 2 > 3 Two plus two is greater than or equal to 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
Writing inequalities
Let's translate each statement into an inequality.
words
x is less than 10
translate to
inequality statement
10
x
<
20 is greater than or equal to y
20
>
y
You try a few:
1. 14 > a
2. b 8
3. 6 < 20f
4. t + 9 36
5. 7 + w 10
6. 19 - p 2
7. n < 12
8. s < 50
9. p > 275
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
Answers
Do you speak math?
Try to change the following expressions from English into math.
Answer
2x 6
Twice a number is at most six.
Answer
2 + x 4
Two plus a number is at least four.
Three less than a number is less than three times that number.
x - 3 < 3x
Answer
The sum of two consecutive numbers is at least thirteen.
Answer
x + (x + 1) 13
Three times a number plus one is at least ten.
3x + 1 > 10
Answer
Solution Sets
A solution to an inequality is NOT a single number. It will have more than one value.
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
This would be read as the solution set is all numbers greater than or equal to negative 5.
Let's name the numbers that are solutions of the given inequality.
r > 10
Which of the following are solutions? {5, 10, 15, 20}
10 > 10 is not true
So, not a solution
5 > 10 is not true
So, not a solution
15 > 10 is true
So, 15 is a solution
20 > 10 is true
So, 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 5d
30 5(5)
30 25
30 5d
30 5(4)
30 20
30 5d
30 5(6)
30 30
30 5d
30 5(7)
30 35
30 5d
30 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 .
Graphing Inequalities
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.
-5
5
-4
-3
-2
-1
1
2
0
3
4
x < 1
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.
-5
5
-4
-3
-2
-1
1
2
0
3
4
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.
-5
5
-4
-3
-2
-1
1
2
0
3
4
Graphing Inequalities
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
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
You try
Graph the inequality
x > 5
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Graph the inequality
-3 > x
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Try these.
Graph the inequalities.
1. x > 4
-5
-4
-3
-2
-1
1
2
5
0
3
4
2. x < -5
-5
-4
-3
-2
-1
1
2
5
0
3
4
Try these.
State the inequality shown.
1.
-5
-4
-3
-2
-1
1
2
5
0
3
4
2.
-5
-4
-3
-2
-1
1
2
5
0
3
4
7
Would this solution set be x > 4?
True
False
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
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 >.
8
-5
5
-4
-3
-2
-1
0
1
2
3
4
A
x > 3
B
x < 3
C
x < 3
D
x > 3
9
5 6 7 8 9 10 11 12 13 14 15
A
11 < x
B
11 > x
C
11 > x
D
11 < x
10
-5
5
-4
-3
-2
-1
0
1
2
3
4
A
x > -1
B
x < -1
C
x < -1
D
x > -1
11
-5
5
-4
-3
-2
-1
0
1
2
3
4
A
-4 < x
B
-4 > x
C
-4 < x
D
-4 > x
12
-5
5
-4
-3
-2
-1
0
1
2
3
4
A
x > 0
B
x < 0
C
x < 0
D
x > 0
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.
An employee earns
e
at least
>
$7.50
7.5
7.5
0 1 2 3 4 5 6 7 8 9 10
Try this:
The speed limit on a road is 55 miles per hour. Define a variable, write an inequality and graph the solution.
Let s = speed
0 < s < 55
Answer
-10 0 10 20 30 40 50 60
13
The sign shown below is posted in front of a roller coaster ride at the
Wadsworth County Fairgrounds. If h represents the height of a rider in inches, what is a correct translation of the statement on this sign?
All riders MUST be
at least 48 inches tall.
A
h < 48
B
h > 48
C
h 48
D
h 48
Simple Inequalities Involving Addition
and Subtraction
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Objectives
Solve one-step inequalities by using addition.
Solve one-step inequalities by using subtraction.
Who remembers how to solve an algebraic equation?
x + 3 = 13
- 3 - 3
x = 10
Use the inverse of addition
Does 10 + 3 = 13
13 = 13
Be sure to check your answer!
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.
To find the solution, isolate the variable x.
Remember, it is isolated when it appears by itself on one side of the equation.
12 > x + 6
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 + 6
12 > 6 + 6
12 > 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.
6 > x
6 > 4
0 1 2 3 4 5 6 7 8 9 10
Solve and graph.
A. k + 3 > -2
k + 3 > -2
- 3
-3
k > -5
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
-5 is not included in solution set; therefore we graph with an open circle.
Solve and graph.
B. r - 9 > 2
r - 9 > 2
+ 9
+9
r > 11
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Solve and graph.
C. 9 > w + 4
9 > w + 4
- 4
- 4
5 > w
w < 5
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
14
Solve the inequality and graph the solution.
n - 2 >
1
2
1
3
5
6
2
A
-5
5
-4
-3
-2
-1
1
2
0
3
4
5
6
2
B
-5
5
-4
-3
-2
-1
1
2
0
3
4
5
6
2
C
-5
5
-4
-3
-2
-1
1
2
0
3
4
5
6
2
D
-5
5
-4
-3
-2
-1
1
2
0
3
4
15
Solve the inequality and graph the solution.
2 < s + 8
A
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
B
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
C
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
D
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
16
Solve the inequality and graph the solution.
-6 + b < -4
A
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
B
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
C
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
D
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
17
Solve the inequality and graph the solution.
-5 > b - 2
A
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
B
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
C
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
D
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
18
Solve the inequality and graph the solution.
3.5 < m + 2
1.5
A
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
1.5
B
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
1.5
C
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
1.5
D
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Simple Inequalities Involving Multiplication
and Division
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Objectives
Solve one-step inequalities by using multiplication.
Solve one-step inequalities by using division.
Multiplying or Dividing by a Positive Number
3x > -27
3x > -27
3 3
x > -9
Since x is multiplied by 3, divide both sides by 3 for the inverse operation.
Remove for Graph
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Solve the inequality and graph the solution.
2
3
r < 6
Since r is multiplied by 2/3,
multiply both sides by the reciprocal of 2/3.
(
)
(
)
3
2
2
3
3
2
r < 6
r < 9
Remove for Graph
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
19
4k > 24
A
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
B
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
C
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
D
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
20
-50 > 5q
A
10 > q
B
-10 < q
-10 > q
C
D
10 < q
21
X
2
< -1
A
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
B
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
C
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
D
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
3
4
22
g > 27
A
g > 36
B
g > 108
g > 36
C
g > 108
D
23
-28 > 4d
A
d > -7
B
d > -7
C
d < -7
D
d < -7
Now let's see what happens when we multiply or divide by negative numbers.
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.
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?
Helpful Hint
The direction of the inequality changes only if the number you are using to multiply or divide by is negative.
Solve and graph.
A.
-3y > 15
-3y< 15
-3 -3
y < -5
Dividing each side by -3 changes the > to <.
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Solve and graph.
B.
7m < 21
7m < 21
7 7
m < 3
Divide each side by 7
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Solve and graph.
C.
5m > -25
5m >-25
5 5
m > -5
Divide each side by 5.
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Solve and graph.
D. -8y > 24
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
E. 9f > 45
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
-r
2
< 5
(
)
(
)
-r
2
Multiply both sides by the reciprocal of -1/2.
-2
> 5
-2
Why did the inequality change?
r > -10
You multiplied by a negative.
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Try these.
Solve and graph each inequality.
1. -7h < 49
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
2. 3x > -15
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Try these.
Solve and graph each inequality.
3. 7m < 21
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
a
-2
4. > -2
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
24
Solve and graph.
2y < -4
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
25
Solve and graph.
x
-1
< -4
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
26
Solve and graph.
-5y -25
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
27
Solve and graph.
n
-2
> 3
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
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
Return to Table of Contents
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.
You can solve two step inequalities in the same way you solve equations.
3x - 10 14
is solved in the same way as
3x - 10 = 14
You can add any positive or negative number to both sides of the inequality.
3x - 10 14
+ 10 +10
3x<24
3 3
x < 8
You can multiply or divide both sides of an equality by any positive number.
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 1: Use additive inverse
Step 2: Use multiplicative inverse
2. Solve this two-step inequality.
26 < 3n + 1
-1 - 1
25<3n
3 3
8 < n
Step 1: Use additive inverse
Step 2: Use multiplicative inverse
1
3
Solve 4p - 9 23
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 }
Move to reveal graph
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Try these.
Solve and graph each inequality.
1. 6 - x > 3
2. -4c + 16 < 0
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Try these.
Solve and graph each inequality.
3. -3y - 21 < 0
4. 22 < -5x + 18x - 4
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Solve and graph the solution.
18 < 4(x + 2)
28
A
2.5 < x
B
2.5 > x
C
2.5 < x
D
2.5 > x
Solve and graph the solution.
16 - x > 7x
29
A
2 < x
B
2 > x
C
2 < x
D
2 > x
Solve and graph the solution.
8 < 5x + 3
30
A
1 < x
B
1 > x
C
1 < x
D
1 > x
Solve and graph the solution.
12 + 5x < 32
31
A
x > 4
B
x < 4
C
x < 4
D
x > 4
Solve and graph the solution.
36 > -3(x - 5)
32
A
-7 < x
B
-7 < x
-7 > x
C
-7 > x
D
33
Which graph represents the solution set for:
1 2 5
2 3 6
Question from ADP Algebra I
End-of-Course Practice Test
<
x
A
-2
2
-1
0
1
B
-2
2
-1
0
1
C
-2
2
-1
0
1
D
-2
2
-1
0
1
Find all negative odd integers that satisfy the following inequality:
3x + 1 17
x + 5 < 17
34
Which value of x is in the solution set of
A
8
B
9
C
12
D
16
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 inequality
2x - 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
x + 3 < 2x - 6
A
x < 5
6
B
x > 5
6
C
x < 6
D
x > 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 1
The cost of 1 DVD and the unknown number of CDs must be less or equal to $65.
Hint 2
How 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
Matt has less than $150, he spent $30 on snacks and bought 3 pairs of pants.
Are thinking about inequalities?
Would you represent the pants or the snacks with a variable?
Hint
Answer
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.
1.Let t = number of tickets
18t + 8 < 60
2.60 - 7w < 15
- 7w < -45
w > 6
Answers
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?
3
7
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.
Let s = minimum score you must get
s + 58 + 59 > 180
s + 117 > 180
- 117 - 117
s > 63
You must score at least 63 to earn an A.
Answer
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
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 Compound
Inequalities
Return to Table of Contents
Objectives
Solve inequalities that contain more than one operation.
Graph solution sets of compound inequalities.
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.
Compound Inequalities
Here are some samples
x > -2 AND x < 3
-2 < x < 3
x -2 AND x 3
-2 x 3
-4
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
4
NOTE: "and" means intersection, so you graph the intersection of the two inequalities
Compound Inequalities
Here are some additional samples
x < -2 OR x > 3
x -2 OR x 3
-4
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
4
NOTE: "or" means union, so you graph the union of the two inequalities
41
Which inequality is represented in the graph below?
5 4 3 2 1 0 1 2 3 4 5
A
4 < x < 2
B
4 x < 2
C
4 < x 2
D
4 x 2
42
Which inequality is represented in the accompanying graph?
3 0 4
A
3 x < 4
B
3 x 4
C
3 < x < 4
D
3 < x 4
Solving Compound Inequalities that contain an AND statement
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.
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
Step 3 Graph your solution
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
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
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.
Pull
3x + 2
"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
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
How would you write this?
x -6 OR x 0
Move to find out
Writing a Compound Inequality From a Graph
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
How would you write this?
-5 < x < 2
Move to find out
Try these.
Solve and graph the solution set.
1. -18 < 3x - 6 < -3
2. -5x + 2 > 27 or x - 3 > 2
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Try these.
Solve and graph the solution set.
3. -2x - 6 > 4 or x + 5 > 8
4. -6 < 2x + 4 < 10
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
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
24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54
B
24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54
C
24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54
D
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
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
B
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
C
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
46
Which graph represents the solution set for 2x - 4 8 and x + 5 7?
A
1 2 3 4
5 67
B
1 2 3 4
5 67
C
1 2 3 4
5 67
D
5 67
1 2 3 4
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
5
3
A
x < 7 or x >
5
3
5
3
B
x < or x >
5
3
C
x < 7 or x <
5
3
D
x < 7 or x >
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.
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
A
x < -5 or x > 1
B
x < -5 and x > 1
C
1 < x and x > -6
D
x > -5 or x > 1
51
Write the inequality shown by the graph.
10
8
9
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
A
x > -7 or x < 3
B
x > -7 and x < 3
C
x > -7 or x > 3
D
x > -7 and x < 3
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
Why is zero a boundary?
It is not possible to use less than zero minutes. Therefore, zero is a second boundary.
About
Think
-100 -50 0 50 100 150 200 250 300 350 400
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
8 < 3x + 2 < 11
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.
Read the problem over and write down which age groups receive a reduced fare.
Do people over 65? Who else?
Hint
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 65
Does 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
Return to Table of Contents
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 and
the 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)