< ≤ > ≥
Basics
An algebraic inequality is a particular type of algebraic expression using inequality signs.
An algebraic inequality is often a word problem.
A word problem requires you to translate Everyday Language into Math Action.
The Everyday Language translated into Math Action in algebraic inequalities includes:
< less than
≤ less than or equal to
> greater than
≥ greater than or equal to
When working with inequalities, it is helpful to know your multiplication and division tables. However, the TI-30XS MultiView Scientific Calculator can be used as needed.
When each side of an algebraic inequality is multiplied by a positive number, the inequality maintains the same direction.
2 < 5
2(2) < 2(5)
4 < 10
5 > 2
3(5) > 3(2)
15 > 6
x + 2 < 5
2(x+ 2) < 2(5)
[There is an invisible multiplication sign between 2 and both components of (x + 2)]
2·x + 2·2 > 2·5
2x + 4 < 10
x + 2 > 5
3(x+ 2) > 3(5)
[There is an invisible multiplication sign between 3 and both components of (x + 2)]
3·x + 3·2 > 3·5
3x + 6 > 15
When each side of an algebraic inequality is multiplied by a negative number, the inequality reverses direction.
2 < 5
-2(2) > -2(5)
-4 > -10
5 > 2
-3(5) < -3(2)
-15 < -6
x + 3 < 5
-2(x+ 3) > -2(5)
[There is an invisible multiplication sign between -2 and both components of (x + 3)]
-2·x + -2·3 > -2·5
-2x + -6 > -10
[ + -6 is the same as – 6 (see negatives) ]
-2x – 6 > -10
x + 2 > 5
-3(x+ 2) < -3(5)
[There is an invisible multiplication sign between -3 and both components of (x + 2)]
-3·x + -3·2 < -3·5
-3x + -6 < -15
[ + -6 is the same as – 6 (see negatives) ]
-3x – 6 < -15
Dictionary – Algebraic Inequalities
The following table lists Math actions under the Everyday words that they represent.
| Less Than | Less Than or Equal To |
Greater Than | Greater Than or Equal To |
| < | ≤ | > | ≥ |
Inequalities |
Dictionary – Algebraic Expressions
The following table lists Everyday words under the Math actions that they represent.
| Addition | Subtraction | Multiplication | Division |
| add | subtract | multiply | divide |
| sum | difference | product | quotient |
| plus | minus | times | share |
| more | less | compound | split |
| increase | decrease | augment | fit |
| greater | fewer | ||
Expressions |
Question
How do you express y is greater than 4?
Answer
y > 4
Question
The number of dogs d in our neighborhood is greater than 7.
Write an inequality that represents our neighborhood’s dog status.
Answer
d > 7
Question
How do you express y is greater than or equal to 4?
Answer
y ≥ 4
Question
The number of dogs d in our neighborhood is greater than or equal to 7.
Write an inequality that represents our neighborhood’s dog status.
Answer
d ≥ 7
Question
How do you express y is less than 4?
Answer
y < 4
Question
The number of cats c in our neighborhood is less than 7.
Write an inequality that represents our neighborhood’s cat status.
Answer
c < 7
Question
How do you express y is less than or equal to 4?
Answer
y ≤ 4
Question
The number of cats c in our neighborhood is less than or equal to 7.
Write an inequality that represents our neighborhood’s cat status.
Answer
c ≤ 7
Question
Seven times the number of birds b in our neighborhood is greater than 1500.
Write an inequality that represents these circumstances.
Answer
7b > 1500
Question
Seven times the number of birds b in our neighborhood is greater than or equal to 1500.
Write an inequality that represents these circumstances.
Answer
7b ≥ 1500
Question
40 more than seven times the number of birds b in our neighborhood is greater than 1500.
Write an inequality that represents these circumstances.
Answer
7b + 40 > 1500
Answer Process
7 times the number of birds: 7b
40 more than 7 times the number of birds: 7b + 40
greater than 1500: > 1500
putting it all together: 7b + 40 > 1500
Question
40 fewer than the number of hamsters h in our neighborhood shared among a total of 7 neighborhoods (including our own) is less than 1500.
Write an inequality that represents these circumstances.
Answer
Answer Process
number of hamsters shared among 7 neighborhoods:
40 fewer than number of hamsters shared among 7 neighborhoods:
less than 1500: < 1500
putting it all together:
Question
40 fewer than the number of hamsters h in our neighborhood shared among a total of 7 neighborhoods (including our own) is less than or equal to 1500.
Write an inequality that represents these circumstances.
Answer
Answer Process
number of hamsters shared among 7 neighborhoods:
40 fewer than number of hamsters shared among 7 neighborhoods:
less than or equal to 1500: ≤ 1500
putting it all together:
Question
A lemonade stand must generate more than $25.00 in order to turn a profit. It spends $10.00 on supplies. It expects 14 customers to buy lemonade. Write an inequality in terms of the amount x that the lemonade stand must charge each customer in order to turn a profit.
Answer
14x + 10 > 25
Answer Process
sales: 14x
sales plus cost of supplies: 14x + 10
more than 25 is the same as greater than 25: > 25
putting it all together: 14x + 10 > 25
Question
A lemonade stand must generate $25.00 or more in order to break even or turn a profit. It spends $10.00 on supplies. It expects 14 customers to buy lemonade. Write an inequality in terms of the amount x that the lemonade stand must charge each customer in order to break even or turn a profit.
Answer
14x + 10 ≥ 25
Answer Process
sales: 14x
sales plus cost of supplies: 14x + 10
25 or more is the same as greater than or equal to 25: ≥ 25
putting it all together: 14x + 10 ≥ 25
Question
Multiply each side of 6 > 4 by 2.
Answer
12 > 8
Answer Process
When each side of an algebraic inequality is multiplied by a positive number, the inequality maintains the same direction.
6 > 4
2(6) > 2(4)
12 > 8
| Input | Display | Comment |
 |
blinker | clears screen |
| 2 × 6 | 2*6 | |
 |
12 | |
| 2 × 4 | 2*4 | |
|
8 | |
Inequality |
Question
Multiply each side of 6 > 4 by -2.
Answer
-12 < -8
Answer Process
When each side of an algebraic inequality is multiplied by a negative number, the inequality reverses direction.
6 > 4
-2(6) < -2(4)
-12 < -8
| Input | Display | Comment |
|
blinker | clears screen |
| -2 × 6 | -2*6 | |
|
-12 | |
| -2 × 4 | -2*4 | |
|
-8 | |
Inequality |
Question
Multiply each side of 14x + 10 > 25 by 2.
Answer
28x + 20 > 50
Answer Process
When each side of an algebraic inequality is multiplied by a positive number, the inequality maintains the same direction.
14x + 10 > 25
2(14x + 10) > 2(25)
[There is an invisible multiplication sign between 2 and both components of (14x + 10)]
2·14x + 2·10 > 2·25
28x + 20 > 50
| Input | Display | Comment |
|
blinker | clears screen |
| 2 × 14 | 2*14 | |
|
28 | |
| 2 × 10 | 2*10 | |
|
20 | |
| 2 × 25 | 2*25 | |
|
50 | |
Inequality |
Question
Multiply each side of 14x + 10 > 25 by -2.
Answer
-28x – 20 < -50
Answer Process
When each side of an algebraic inequality is multiplied by a negative number, the inequality reverses direction.
14x + 10 > 25
-2(14x + 10) < -2(25)
[There is an invisible multiplication sign between -2 and both components of (14x + 10)]
-2·14x + -2·10 < -2·25
-28x + -20 < -50
[ + -20 is the same as – 20 (see negatives) ]
-28x – 20 < -50
| Input | Display | Comment |
|
blinker | clears screen |
| -2 × 14 | -2*14 | |
|
-28 | |
| -2 × 10 | -2*10 | |
|
-20 | |
| -2 × 25 | -2*25 | |
|
-50 | |
Inequality |
Practice – Questions
1. How do you express y is less than 9?
2. How do you express y is greater than or equal to 9?
3. Multiply each side of 5 < 7 by 3.
4. Multiply each side 11x + 8 > 13 by -3.
5. The number of gerbils g in our neighborhood is greater than or equal to 19.
Write an inequality that represents our neighborhood’s gerbil status.
6. 77 more than 9 times the number of gerbils g in our neighborhood is greater than 1300. Write an inequality that represents these circumstances.
7. A chili supper is being organized to raise money for charity. The food and banquet room cost $350.00. The dinner guests g will be charged $5.00 apiece. Write an inequality in terms of g that shows the amount of money the chili supper needs to generate to break even or turn a profit.
8. A restaurant must generate more than $5000.00 a day in sales in order to turn a profit. It spends $3000.00 a day on wages and supplies. It expects 44 customers a day. Write an inequality in terms of the the amount x the restaurant must charge each customer in order to turn a profit.
9. A shopper has a budget to spend less than or equal to $75.00 on paperback books ordered from an online vendor. Each book costs $9.99. Regardless of how many books b are ordered, shipping and handling costs $7.95. Write an inequality in terms of b that keeps the shopper within the specified budget.
10. A golfer has a budget to spend less than $15,000.00 a year on his hobby. The country club charges an annual fee of $10,000.00 plus $50.00 for every round r of golf played.
Write an inequality in terms of r that keeps the golfer within the specified budget.
Practice – Answers
1. y < 9
2. y ≥ 9
3. 15 < 21
4. -33x – 24 < -39
5. g ≥ 19
6. 9g + 77 > 1300
7. 5g ≥ 350
8. 44x + 3000 > 5000
9. 9.99b + 7.95 ≤ 75.00
10. 50r + 10000 < 15000