Equations and Proportions Notes Module 2 Equations and Proportions

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)

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

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 -3 -3 3x = 15 3 3 Term 1: x  5 Term 2: x + 1  5 + 1 = 6 Term 3: x + 2  5 + 2 = 7 Answer: 5, 6, 7 x = 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 +3 +3 |x + 4| = 13  x + 4 = 13 & x + 4 = -13 -4 -4 -4 -4 x = 9 & x = -17 Note: Absolute values can never equal a negative number |x| = -(#) then no solution

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

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

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

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’

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