### Inequality Notes

```INEQUALITIES
an Inequality?
> 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.
“nine is greater than
ex: 9 > 3n is read
three times a number”
< 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.
“four times a number
ex: 4v < 19 is read is less than nineteen”
an Inequality?
≥ 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 “three times a number is greater
than or equal to nine”
≤ 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.
“four times a number is less than or
ex: 4x ≤ 24 is read
equal to twenty-four”
an Inequality?
Examples:
fourteen is greater than two times a number
a) 14 > 2n ______________________________________
fifteen is greater than or equal to three
plus a number
b) 15 ≥ 3 + n ____________________________________
twenty-three is less than or equal to
thirty minus a number
c) 23 ≤ 30 – n ___________________________________
twelve is less than a number divided by
nine
d) 12 < n ÷ 9 ___________________________________
nine plus a number is less than twentye) 9 + n < 24 – n four
________________________________
minus that same number.
Is It
a
Solution?
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?
No, if you plug in 10 for n then you get “19 is greater than 22”
which is a false statement.
ex 2: If b = 15 is
b
3
+ 8 ≥ 13 a true 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.
Is It
a
Solution?
Which of the following inequalities is/are incorrect when n = 10?
a) 6n + 8 > 68 No, if you plug in 10 for n then you get “68 is
greater than 68” which is a false statement.
b) 5(6 + n) ≥ 80
c)
n
5
d)
2( n  15)
5
– 10 < 0
≤9
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
a) Is more than
b) Is greater than
c) Is larger than
d) Is above
e) Is bigger than
Greater Than or Equal To
a) Is at least
b) Is no less than
c) Is no smaller than
d) Is at minimum
Less Than
a) Is less than
b) Is smaller than
c) Is below
Less Than or Equal To
a) Is at most
b) Is no more than
c) Is no greater than
d) Is at maximum
Translating
Inequalities
5>n
a) five is greater than a number: _____
16 ≤ n
b) sixteen is less than or equal to a number: ______________
c) two times the difference of a number and four is greater
2n – 4 ≥ 75
than or equal to seventy five: ______________
d) fourteen less than a number is at least seventeen:
n – 14 ≥ 17
_____________
n
e) the difference of half a number
 7  18
2
and seven is no more than eighteen: ______________
Is It
a
Solution?
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. 16n ≥ 120
16
n
Yes,IfCarl
n = 8,
willwill
need
Carl128
ft. whicha isdiscount?
more than
the required
Explain
120 sq. ft.
Is It
a
Solution?
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.
let m = months
500 – 75m > 200
b) December is 6 months away… will he meet his goal?
Explain.
No, when we substitute 6 in for m we have an incorrect inequality.
It reads “50 is greater than 200” which is not true.
Is It
a
Solution?
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.
let p = pounds
53.25 + p ≤ 82.5
b) If Kelly collects an additional 29.1 pounds of food will she
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.
Is It
a
Solution?
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.
let m = Martina’s last test score
72 + 91 + 84 + 82 + m ≥ 425
b) If Martina earns a 96% on her next test will she meet her
goal? Explain.
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.
Graphing
Inequalities
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.
Graphing
Inequalities
Is 9 a solution to the inequality represented
by the graph?
YES
NO
What is the inequality this graph represents? n ≥ -7
Graphing
Inequalities
Is -6 a solution to the inequality represented
by the graph?
YES
NO
What is the inequality this graph represents? n ≥ -3
Graphing
Inequalities
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”
What is the inequality this graph represents? n < 5
Solving
Inequalities?
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?