Inequalities

Scott Solomon

<                              ≤                              >                              ≥

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
\bf\displaystyle\frac{h}{7} – 40 < 1500

Answer Process
number of hamsters shared among 7 neighborhoods: \bf\displaystyle\frac{h}{7}
40 fewer than number of hamsters shared among 7 neighborhoods:  \bf\displaystyle\frac{h}{7} – 40
less than 1500:  < 1500
putting it all together:  \bf\displaystyle\frac{h}{7} – 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
\bf\displaystyle\frac{h}{7} – 40 ≤ 1500

Answer Process
number of hamsters shared among 7 neighborhoods: \bf\displaystyle\frac{h}{7}
40 fewer than number of hamsters shared among 7 neighborhoods:  \bf\displaystyle\frac{h}{7} – 40
less than or equal to 1500:  ≤ 1500
putting it all together:  \bf\displaystyle\frac{h}{7} – 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
clear blinker clears screen
2 × 6 2*6
enter 12
2 × 4 2*4
enter 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
clear blinker clears screen
-2 × 6 -2*6
enter -12
-2 × 4 -2*4
enter -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
clear blinker clears screen
2 × 14 2*14
enter 28
2 × 10 2*10
enter 20
2 × 25 2*25
enter 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
clear blinker clears screen
-2 × 14 -2*14
enter -28
-2 × 10 -2*10
enter -20
-2 × 25 -2*25
enter -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

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