Pre Algebra (Period 2 & 4)

Graphing & Writing Inequalities Review

Some words to really know:
At least--> means greater than or EQUAL TO

I want at least $150 to go shopping for holiday gifts!!
m ≥ 150


AT MOST --> less than or EQuAL TO

You might say that you want at most 15 minutes of homework tonight. That means you will take 15 minutes but you sure would like less....
m ≤ 15

Review of some of the most missed problems from our recent races:
y is at most 5
y ≤ 5

x is no more than 7
x ≤ 7

y is at least 20

y ≥ 20

y is less than 8
y < 8. Now 5 < y reads " Five is less than than y" --> if that is true then y must be greater than 5 or

y > 5 so now you can graph easily because you know exactly which way the arrow ( solution set) must be pointing... in the same direction as the inequality symbol. THis ONLY works when you have the variable on the left!! as in y > 5.
ALWAYS CHANGE so the variable is on the left!!

Solve one step inequalities just like you do for equations.. just make sure to end with the variable on the left-- to be able to graph easily

y - 7 > -3
add 7 to BOTH Sides
y > 4
Now you can graph easily... It's an open dot on your graph Because it DOES NOT inlcude 4 as part of the solution

FORMAL CHECK
(1) REWRITE the inequality
(2) Substitute in for the variable--using ANY number that fits the solution (except the boundary number) In this example y > 4,
4 is the boundary number. We showed in class why you can't use the boundary number so if y > 4 then 5, 10, 100, etc would work...
In class we used 10
10 -7 > -3 (Remember to put a "?" over the inequality)
(3) REALLY DO THE MATH. that is, in this case,
do 10-7.
Well 10-7 = 3 so
3 > -3 That's true-- it works!!

If you get something that is not true, chances are you did your problem wrong. re work the problem and see if you can discover your error!!

WHen dividing or multiplying by a negative coefficient (that's the number attached to the variable) you must remember to
REWRITE the inequality AS YOU "FLIP THE SWITCH" (change the inequality symbol)
y/-5 ≥ -3
We need to multiply both sides by a -5 so we must REWRITE and FLIP
(-5)(y/-5) ≤ (-3)(-5)
y ≤ 15

FORMAL CHECK
(1) REWRITE THE INEQUALITY
(2) Substitute in one of the SOLUTIONS-- any one that really fits and is easy to work with!!
(3) DO THE MATH-- look at your results-- does it make sense?... don't just put a happy face!!

(1) y/-5 ≥ -3
well, we found out the y ≤ 15 that means any number less than 15 could be used to check I would pick something that was easy to work with, something that was divisible by 5... like 0, or 5 or even 10
we used 0 in class
(2) 0/-5 ≥ -3
(3) 0 ≥ -3 That's true!! IT WORKED

But what if I used 5?
well step 2 would be
(2) 5/-5 ≥ -3
(3) -1 ≥ -3 and that's still true !!


5y ≤ -35
I just divide by 5. I don't need to switch anything because I am NOT dividing by a negative number
5y/5 ≤ -35/5
y ≤ -7

But what about
-15 ≤ -3m

In this case I must rewrite as I "FLIP THE SWITCH" (Remember the poster from class)
-15/-3 ≥ -3m/-3
5 ≥ m

Before we continue we need to realize that if five is greater than or equal to m... that means m must be less than or equal to 5 or m ≤ 5. Now its easy to graph.

FORMAL CHECK
-15 ≤ -3m
-15 ≤ -3(4)
-15 ≤ -12 which is true!!

In class we used a number that was not correct to show what happens.

When working with Inequalities with 2 steps and variables on both sides:
(1) Do Distributive Property (DP) carefully-- if necessary
(2) combine like terms on each side
(3) "Jump" the variables to one side of the equation
(4) Add or subtract
(5) Multiply or Divide-- being careful if have negative coefficients to rewrite the inequality and "flip the Switch"
(6) MAKE SURE the variable is on the LEFT Side