**< ≤ > ≥**

** 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 orEqual To |
Greater Than |
Greater Than orEqual 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
** – 40 < 1500

**Answer Process
**number of hamsters

*shared*among 7 neighborhoods:

40

*fewer*than number of hamsters shared among 7 neighborhoods: – 40

*less than*1500: < 1500

putting it all together: – 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 or equal to 1500.

Write an inequality that represents these circumstances.

**Answer
** – 40 ≤ 1500

**Answer Process
**number of hamsters

*shared*among 7 neighborhoods:

40

*fewer*than number of hamsters shared among 7 neighborhoods: – 40

*less than or equal to*1500: ≤ 1500

putting it all together: – 40 ≤ 1500

**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