Math 6H Period 3, 6 & 7

Solving Equations & Inequalities 2-4 & 2-5
Any value of the variable that makes the equation or inequality a true sentence is called a solution. We solve an equation or inequality when we find ALL its solutions.

Remember when you solved the following:
? + 4 = 12.
Your teacher would ask you what number replaced the ?
We are going to learn how to solve equations and inequalities using Algebraic terms-- we will solve using inverse operations and we will justify our steps.
x - 4 = 12
add 4 TO BOTH SIDES of the equation using
the addition property of equality ( + prop =)
so
x - 4 = 12
+ 4 = + 4
x + 0 = 16 but the Identity Property of addition ( ID+) allows us to write
x = 16

Similarly
15n = 60
can be read as "fifteen times n equals sixty"
What undoes multiplication? What is the inverse operation of multiplication?
DIVISION
so we will divide both sides by 15 to isolate the variable.
15n = 60
15 = 15
using the Division Property of Equality ÷prop +
Then we get
1n = 4
but the Identity Property of Multiplication ( ID X) lets us write
n = 4
See the Class Notes Solving One-Step Equation ( Yellow Sheet) for more examples. Make sure to glue that into your spiral notebook.



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.