INEQUALITIES
Consider the following scenario….You want to save enough money to buy yourself that “thing” you always wanted….The
“thing” costs $275 so you volunteer to do chores around the house for $$$. Your parents decide to pay you $3.50 per
chore and as an added incentive they have given you 40 bucks.
How many chores would you need to do to have at least the $275?
Below are four possible equations/inequalities…Which one seem to apply in the problem above?
Assume x = number of chores you need to do
a) x + 40 = 275
b) 3.50x + 40 < 275
c) 3.50x + 40 = 275
d) 3.50x + 40 ≥ 275
So when a situation calls for “at least” or “at most” we will need to set up an inequality (that means greater than or less
than)
Recall:
At least means that many or more which is the symbol ≥
At most means that many or less which is the symbol ≤
The symbol > means more than and the symbol < means less than
So how do you solve an inequality?
Well for the most part they are solved exactly like equations (get the variable expressions on one side and the numbers
on the other, then divide by the coefficient of the variable) there are a few exceptions (which we will discuss soon)
So for now solve the inequality that describes the situation above to determine how many chores you would need to do
3.50x + 40 ≥ 275
You could represent that solution multiple ways…ALGEBRAICALLY, GRAPHICALLY, IN SET NOTATION
Now think about this situation
3 is less than 10 which symbolically looks like this
3<10
Now suppose you added 20 to both sides, is it still true?
Now suppose you went back to the original 3<10
Now suppose we multiplied both sides by 5, is it still true?
Now suppose you went back to the original 3<10
Now suppose we multiplied both sides by -8, is it still true?
So the first difference between equations and inequalities is multiplying (or dividing) by negatives…..when you do so
with an inequality, you need to switch the inequality symbol….
Here is an example of me using that idea
12 – 3x < 48
-3x < 36
x > -12 notice in the last step I divided by -3 so I flipped the inequality
I could also represent that solution set graphically on a number line like this
*use open circle if the solution does not include that actual number and fill in the circle if the solution does include that
actual number
Now to check it I could take any number in the solution set, any number larger than -12 and see if it makes a true
statement from the original….I will pick 100, since 100 is clearly larger than -12
12 – 3(100) < 48
12 – 300 < 48
-288 < 48 yep it’s true so I am confident that my solution is correct
Solve each inequality given and show the solution set ALGEBRAICALLY, GRAPHICALLY on a number line and in SET
NOTATION
13𝑥 < 9(1 − 𝑥)
−5(𝑥 − 1) ≥ 10
5𝑥−6
3(x – 6) > x + 12
2𝑥+1
≥
−15𝑥 ≥ −45
3
3−𝑥
2
7
≤5+𝑥
*when cross multiplying with an inequality you must do top left times bottom right first and put that on the left side of
the inequality, then do bottom left times top right and put that on the right of the symbol…this should make sense if you
look at a basic inequality like
1
4
3
< 4 therefor 4<12.
Take some time to check your solutions
Thinking about two inequalities at one time or in other words COMPOUND inequalities
Graph the solution set to each compound inequality on a number line.
e.
𝑥 < −8 or 𝑥 > −8
f.
0 < 𝑥 ≤ 10
Write a compound inequality for each graph.
g.
h.
i.
A poll shows that a candidate is projected to receive 57% of the votes. If the margin for error is plus or minus 𝟑%, write a
compound inequality for the percentage of votes the candidate can expect to get.
j.
Mercury is one of only two elements that is liquid at room temperature. Mercury is non-liquid for temperatures less than
−𝟑𝟖. 𝟎°F or greater than 𝟔𝟕𝟑. 𝟖°F. Write a compound inequality for the temperatures at which mercury is non-liquid.