1. Equations and Proportions
Notes Module 2
Equations and Proportions

2. Notes: Module 2.01 – One Step Equations
1st  Simplify each side: distribute & combine like terms
2nd  If you have variables on both sides, you must add or subtract one of them to move it so you will only have a variable on one side
3rd  Focus on the side with the variable. Add or subtract the number without the variable
4th  Divide or multiply by the coefficient to get the variable alone
Note: the variable can NOT be negative (you have to multiply by -1 on both sides to get rid of the negative)

3. Examples 3x – 5 = 7 +5 +5 3x = 12 3 3 x = 4 6x + 23 = 27 – 2x
8x = 4
x = 4/8 or ½
2 (x + 8) – 9 = 5 2x + 16 – 9 = 5 2x + 7 = x = x = -1

4. Notes: Module 2.02 – Multi-Step Equations
Consecutive Integers Problems
Regular consecutive integers
Odd or Even Consecutive Integers
Term number
How to solve 1 X 2
X + 1 3
X + 2 4
X + 3
Term number
How to solve 1 X 2
X + 2 3
X + 4 4
X + 6
Example: Find three consecutive integers whose sum is 18.
(x) + (x + 1) + (x + 2) = 18
3x + 3 = 18
3x = 15
Term 1: x  5
Term 2: x + 1  = 6
Term 3: x + 2  = 7
Answer: 5, 6, 7
x = 5

5. Notes: Module 2.02 – Multi-Step Equations
Absolute Value Equations
Will always have 2 answers!
1st  Get the absolute value info on one side and everything else on the other
2nd  Set up two equations, each with the expression from the absolute value bars. Set one equal to the original number, and set the other equal to the negative (or opposite) of the original and solve both equations.
Example: |x + 4| - 3 = 10
|x + 4| = 13  x + 4 = 13 & x + 4 = -13
x = 9 & x = -17
Note: Absolute values can never equal a negative number
|x| = -(#) then no solution

6. Notes: Module 2.03 – Literal Equations
Solving for a specific variable
These equations have several variables, but you only focus on the one they tell you. Move the others just how you would (opposite PEMDAS), you just don’t actually work out and solve for a number.
Examples:
Solve for ‘r’ 4qr+2t=7u
-2t -2t
4qr=7u-2t
4q 4q
Solve for ‘y’ xy+w=p
-w -w
xy=p-w
x x
 r = 7u-2t 4q
 y= p-w x

7. Notes: Module 2.05 – Problem Solving
In order to solve word problems, you need to use one of several tools:
Make a table to organize the information
Create and equation and solve
Figure out which value is changing and which value is constant
Make sure you write the equation with a variable AND solve for the variable

8. Notes: Module 2.06 – Proportions
Proportion is two ratios that equal each other (remember: you can’t have a zero in the denominator of either ratio)
In a proportion, the cross products are equal
Percent problems are always out of 100 (meaning the percent is the numerator and 100 will always be the denominator)
Percent of increase or decrease

9. Notes: Module 2.07 – Solving Inequalities
Inequalities usually have a solution that is more than one number
Closed circle (≥ or ≤) means the number is included (greater/less than or equal to)
Open circle (> or <) means the number is NOT included (greater/less than
Solve an equality the same you would solve an equation, except when multiplying or dividing by a negative number **you flip the inequality sign**
‘And’ problems find the solutions that are in common; solutions should go toward each other and stop at each end
‘Or’ problems find what the solutions are for either inequality; usually going away from each other, but if going towards each other, the solution is ‘all real numbers’

10. Notes: Module 2.07 – Solving Inequalities (con.)
Absolute value inequalities