Math 6H Period 3, 6 & 7

Solving Equations & Inequalities 2-4 & 2-5

Inequalities Continued

We know about > greater and as well as < less than
so now we look at
≥ which means " greater than or equal to" and
≤ which means " less than or equal to"

This time the boundary point ( or endpoint) is included in the solution set.
The good news is that we still solve these inequalities the same way in which we solved equations-- using the properties of equality.
w/4 ≥ 3
we multiply both sides by 4/1
(4/1)(w/4) ≥ 3(4/1) by the X prop =
1w ≥ 12
w ≥ 12 by the ID(x)

Which means that any number greater than 12 is part of the solution AND 12 is also part of that solution

Take the following:
b - 3 ≤ 150 ( we need to add 3 to both sides of the equation)
+3 = +3 using the + prop =
b + 0 ≤ 153
or b ≤ 153 using the ID (+)

Problem Solving: Using Mathematical Expression 2-6

Seventeen less than a number is fifty six

I suggest lining up and placing the "equal sign" right under the word is
then complete the right side = 56
after that take your time translating the left
Start with a "let statement."
A "let statement" tells your reader what variable you are going to use to represent the number in your equation.
So in this case Let b = the number
it becomes
b- 17 = 56
Now solve as we have been practicing for a couple of weeks.
b - 17 = 56
+17 = +17 using the +prop=
b + 0 = 73
b = 73 by the ID(+)

How could we check?
A FORMAL CHECK involves three steps:
1) Re write the equation ( from the original source)
2) substitute your solution or... "plug it in, plug it in...."
3) DO the MATH!! actually do the math to check!!

so to check the above
b - 17 = 56
substitute 73 and put a "?" above the equal sign...

73 - 17 ?=? 56
Now really do the math!! Use a side bar to DO the MATH!!
That is, what is 73- 17? it is 56
so 56 = 56


Practice some of the class exercised on Page 50, Just practice setting up the equations from the verbal sentences.

Algebra Period 4

The Multiplication Property of Inequality 4-3
Solving Inequalities  with multiplication or division:
Again, you will use your equation skills, but this time with the Multiplicative Inverse Property.

ONE MAJOR DIFFERENCE FROM EQUATIONS;

When you multiply or divide by a NEGATIVE to BALANCE, you must SWITCH the inequality SYMBOL! 

(REMEMBER --->Does not apply to adding or subtracting negatives.)


EXAMPLE: -3y > 9

You need to divide both sides by NEGATIVE 3 so the symbol will switch from > to < in the solution y < -3 is the answer

If you want to understand why:

3 < 10

Now multiply both sides by -1 (mult prop of equality)

You get -3 < -10, but THAT'S NOT TRUE!!!

You have to SWITCH THE SYMBOL to make the answer true: -3 > -10


REMEMBER: when you MULTIPLY or DIVIDE by a NEGATIVE, the symbol SWITCHES



Using the Properties Together 4-4


Same as equations except make sure you switch the symbol if you multiply or divide by a negative!

Always finish with the variable on the left-- makes it so much easier to graph

Check with whatever solution is easiest in the solution set!

Somethings to remember with two step inequalities:

Before you start, you may want to clear fractions or decimals, but if you don't mind using them, just get started with the checklist below.
If you want to clear them, you should do that right after you distribute (between steps 1 and 2 below)

1. Do distributive property first (if necessary)

2 Combine like terms on each side of the wall (equal sign)

3. Jump the variables to one side of the wall (get all the variables on one side of the equation) by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term)

4. Add or subtract

5. Multiply or divide

6. Make sure the variable is on the LEFT side when finished.

Algebra Period 4

Inequalities & Their Graphs 4-1
Introduction to INEQUALITIES and graphing them:

Writing an inequality

Graphing an inequality - open dot is < or >

Closed dot mean less than or EQUAL or greater than or EQUAL
(think of the = sign as a crayon that you can use to COLOR IN THE DOT!)

Different from equations: Inequalities have many solutions (most of the time an infinite number!)

Example: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)

n ≥ 3 means still means that every real number greater than 3 is a solution, but now 3 is also a solution.
The endpoint ( in this case 3) is called the boundary point.


GRAPHING INEQUALITIES:

First, graphing an equation's solution is easy

1) Let's say you solved an inequality and you discovered that y = 5, you would just put a dot on 5 on the number line

2) But now you have the y ≥ 5

You still put the dot but now also darken in an arrow going to the right

showing all those numbers are also solutions

3) Finally, you find in another example that y > 5

You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!


TRANSLATING WORDS:

Some key words to know
AT LEAST means greater than or equal

AT MOST means less than or equal


I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!

I want at most 15 minutes of homework means that I can have 15 minutes,
but I'm hoping for even less!



The Addition Property of Inequalities 4-2

Solving Inequalities with adding or subtracting

Simply use the Additive Inverse Property as if you were balancing an equation!

The only difference is that now you have more than one possible answer.

Example: 5y + 4 > 29

You would -4 from each side, then divide by 5 on each side and get:
y > 5

Your answer is infinite! Any real number bigger than 5 will work!
It is IMPOSSIBLE to check every value in an infinite solution set. However, once you have determined the endpoint-- or boundary point, you can verify by checking a representative or sample number from the supposed solution set to verify that your inequality has been solved correctly.

Math 6H ( Periods 3, 6, & 7)

Solving Equations & Inequalities 2-4 & 2-5

GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation


Here are the steps and justifications (reasons)

1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)

2. What is the operation the variable is doing with that number in its way?

3. Get rid of that number by using the opposite (inverse) operation

*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem

GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!

4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?

that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:

" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality

5. You should now have the variable all alone (isolated) on one side of the equal sign.

6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]

For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]

7. Put answer in the final form of x = ____and box this in.