Inequalities

INEQUALITIES

> is the “greater than” symbol. It signifies that the value on the left hand side of the inequality is larger than the value on the right hand side of the inequality.

ex: 9 > 3n is read

< is the “less than” symbol. It signifies that the value on the left hand side of the inequality is smaller than the value on the right hand side of the inequality.

ex: 4v < 19 is read

How Do You Read an Inequality?

“nine is greater than three times a number”

“four times a number is less than nineteen”


≥ is the “greater than or equal to” symbol. It signifies that the value on the left hand side of the inequality is equal to or larger than the value on the right hand side of the inequality.

ex: 3x ≥ 9is read

≤ is the “less than or equal to” symbol. It signifies that the value on the left hand side of the inequality is equal to or smaller than the value on the right hand side of the inequality.

ex: 4x ≤ 24 is read

“three times a number is greater than or equal to nine”

“four times a number is less than or equal to twenty-four”


Examples:

a) 14 > 2n

______________________________________

b) 15 ≥ 3 + n

____________________________________

c) 23 ≤ 30 – n

___________________________________

d) 12 < n ÷ 9

___________________________________

e) 9 + n < 24 – n

________________________________

fourteen is greater than two times a numberfifteen is greater than or

equal to three plus a numbertwenty-three is less than or equal to thirty minus a number

nine plus a number is less than twenty-four minus that same number.

twelve is less than a number divided by nine

When given an inequality that contains a variable you can determine if a given solution is true or false by simply substituting the value in the inequality.

ex 1: Is 9 + n > 22 when n = 10?

ex 2: If b = 15 is + 8 ≥ 13 a true statement?

Is It a Solution?

3

b

No, if you plug in 10 for n then you get “19 is greater than 22” which is a false statement.

Yes, if you plug in 15 for n then you get “13 is greater than or equal to 13” which is a true statement.

Which of the following inequalities is/are incorrect when n = 10?

a) 6n + 8 > 68

b) 5(6 + n) ≥ 80

c) – 10 < 0

d) ≤ 9

Is It a Solution?

5

n

5

)15(2 n

No, if you plug in 10 for n then you get “68 is greater than 68” which is a false statement.

No, if you plug in 10 for n then you get “10 is less than or equal to 9” which is a false statement

Which Symbol Should Be Used?

Greater Than Less Than

Greater Than or Equal To

Less Than or Equal To

a)Is more thanb)Is greater thanc)Is larger thand)Is above e)Is bigger than

a)Is less thanb)Is smaller thanc)Is below

a)Is at leastb)Is no less thanc)Is no smaller thand)Is at minimum

a)Is at mostb)Is no more thanc)Is no greater thand)Is at maximum

a) five is greater than a number: _____

b) sixteen is less than or equal to a number: ______________

c) two times the difference of a number and four is greater than or equal to seventy five: ______________

d) fourteen less than a number is at least seventeen: _____________

e) the difference of half a number and seven is no more than eighteen: ______________

Translating Inequalities

5 > n

16 ≤ n

2(n – 4) ≥ 75

n – 14 ≥ 17

1872

n

INEQUALITIES

The lunch lady records the number of sandwiches sold in

the school cafeteria each day. If she sells more than 50 peanut butter sandwiches, she orders

more peanut butter. If s represents the number of sandwiches sold, write an

inequality for this scenario.

S > 50

Mrs. Stanfield must have a minimum of 10 fish in her fish tank (at all times) in the front of the Intermediate Building. If f represents the number of

fish, write an inequality to represent this scenario.

F ≥ 10

In NYC, the elevator ride to the top of the empire state

building can hold no more than 3500 pounds. If p

represents the number of pounds, write an inequality to

represent this scenario.

P ≤ 3500

The gym class’s average results for girls participating in the long jump

is 100 inches.

Sue could jump no farther than 4 inches more than the average

distance for girls. If j represents the distance Sue could jump, write an inequality to represent how far

Sue jumped.

j ≤ 8ft 8inches

Aussie has a minimum of 5 pairs of UGGs boots in her

closet at all times. If u represents the number of UGGs in her closet, write

an inequality for this scenario.

u ≥ 5

Jimmy just bought a new refrigerator on BLACK

FRIDAY. It has a special meat drawer that can hold

no more than 10 lbs of meat. Write an inequality

for this scenario.

Let m = pounds of meat

m ≤ 10

George the elf has c candy canes. George got into a scuffle

with his sister. He ended up breaking 2 of the candy canes.

He needs a minimum of 25 unbroken candy canes for his

Christmas party. Write an inequality for the scenario.

Let c = unbroken candy canes

c – 2 ≥ 25

Julia has $20 to spend on ice cream. Each ice cream cone is $2.75. Julia buys ice cream for

her friends, f. She wants to have a minimum of $10 left in her wallet after she buys all of her friends ice cream. Write an

inequality for this scenario.

20 – 2.75f ≥ 10

Evergreen Farms are selling Christmas trees. They must sell

at least 100 trees to stay in business this year. The first

week they sold 30 trees. The second week they sold 60 trees. The next week they sell t trees. Write an inequality to represent

this situation.

Let c = trees sold

90+ c ≥ 100

Yes they will meet their goal

If the Evergreen Farms sells 15

trees will they stay in business?

Yes they will meet their goal and exceed it by

5 trees

Carl is measuring a room in his home that he needs to purchase new carpeting for. A diagram of the room is shown below. The local carpeting store currently is offering a 20% discount if you purchase at least 120 sq. ft. of carpeting. If the

formula for determining the area of a rectangle is A = bh write an inequality to represent the area

of Carl’s room if he hopes to receive the discount.

Is It a Solution?

n

16 If n = 8, will Carl

receive a discount? Explain

Yes, Carl will need 128 sq. ft. which is

more than the required 120 sq. ft.

16n ≥ 120

Keith has $500 in his savings account at the beginning of June. He wants to have more than $200 in the account on December 1st so that he

can purchase holiday gifts for his family and friends.

Keith withdraws $75 each month for his own expenses.

a) Write an inequality to represent Keith’s situation.

b) December is 6 months away… will he meet his goal? Explain.

Is It a Solution?

let m = months 500 – 75m > 200

No, when we substitute 6 in for m we have an incorrect inequality. It reads “50 is greater than 200” which is not true.

Kelly is collecting canned foods to contribute to the Thanksgiving food drive at her Church. The box Kelly is placing the cans in can hold a maximum of 82.5 pounds. With a month until she has to turn in the

canned goods Kelly has collected 53.25 pounds worth of canned goods. Kelly plans to collect as many more

cans as she can but is not sure if she will need an additional box.

a) Write an inequality to represent Kelly’s situation.

b) If Kelly collects an additional 29.1 pounds of food will she need an additional box?

Is It a Solution?

let p = pounds 53.25 + p ≤ 82.5

No, when we substitute 29.1 in for p we find that “82.35 is less than or equal to 82.5” which is a true statement.

So far this year Martina has earned test scores of 72%, 91%, 84%, and 82%. Martina would like her average to be at least an 85%. To earn this grade the sum of her tests must be a minimum of 425.

a) Write an inequality that represents Martina’s situation.

b) If Martina earns a 96% on her next test will she meet her goal? Explain.

Is It a Solution?

let m = Martina’s last test score 72 + 91 + 84 + 82 + m ≥ 425

Yes, when we substitute 96 in for m we have an inequality that reads “425 is greater than or equal to 425.” This is a true statement.

You can use a number line to represent inequalities.

Step 1: If you are not given a number line that is already numbered, draw a number line that contains the number given in the inequality, a number bigger than the number given, and a number smaller than the number given.

Step 2: Place a dot on the number line For ≤ and ≥ use a closed circle on the number line

o For < and > use an open circle on the number line

Step 3: Draw a dark line on the number line in the direction of the numbers that make the inequality true. Place an arrow on the line you drew to signify that the line continues forever in that direction.

Graphing Inequalities

Examples:

a) n > 4

b) h ≤ 16

c) k ≥ -3

d) n < -17


You can also use the number line to

determine if a value is a solution to an inequality. Simply look to see if the value given is part of the

shaded region; if it is, then it is a solution. If it

happens to be where the dot was drawn, then look to see if the dot is open

or closed. An closed circle means that value is a solution… an open circle means it is not.


Is 9 a solution to the inequality represented by the graph?YES NO

What is the inequality this graph represents?

n ≥ -7

If you were to continue the number line it would extend to the 9 and

therefore be a solution to the inequality


Is -6 a solution to the inequality represented by the graph?YES NO


n ≥ -3

The graph of this inequality will

never pass through -6 on the

number line The inequality represented on this

graph is n ≥ -3 and -6 is not greater than or equal

to -3


Is 5 a solution to the inequality represented by the graph?YES NO

If the circle was closed, then yes 5 would be a

solution. An open circle means that it is simply “less than” in this case, not “less than or EQUAL

to 5”


n < 5

Recently, you learned how to solve equations such as…

3x + 4 = 10 5 + 2n = 75 5v – 18 = 7

To solve an inequality, follow the same exact process as you did for the equations above. This time instead of having the equals (=), leave the inequality symbol (<, >, ≤, or ≥).

When you have finished solving the inequality, graph your solution.

To check, pick a number that is on the line you drew on the graph and substitute that value into the original inequality. Make certain that you now have a true statement!

Solving Inequalities?

Solve each inequality and graph your solution. Then use your graph to check if your solution is accurate.

a) 9 + x > 81

b) 3p – 25 < 14

c) 2.6565c + 6.2 ≥ 16.826

d) .

Solving Inequalities?

11713

w

Documents

Inequalities