2.1 The Addition Property of Equality A linear equation in one variable can be written in the form: Ax + B = 0 Linear equations are solved by getting “x” by itself on one side of the equation Addition Property of Equality: if A=B then A+C=B+C General rule: Whatever you do to one side of the equation, you have to do the same thing to the other side.

2.1 The Addition Property of Equality Example of Addition Property: x – 5 = 12 (add 5 to both sides) x = 17 Example using subtraction:

2.2 The Multiplication Property of Equality Multiplication Property of Equality: if A=B and C is non-zero, then AC=BC (both equations have the same solution set) Since division is the same as multiplying by the reciprocal, you can also divide each side by a number. General rule: Whatever you do to one side of the equation, you have to do the same thing to the other side.

2.2 The Multiplication Property of Equality Example of Multiplication Property: Example using division 5x = 60 (divide both sides by 5) x = 12

2.3 More on Solving Linear Equations: Terms - Review As with expressions, a mathematical equation is split up into terms by the +/-/= sign: Remember, if the +/- sign is in parenthesis, it doesn’t count:

2.3 More on Solving Linear Equations: Multiplying Both Sides by a Number Multiply each term by the number (using the distributive property). Within each term, multiply only one factor. Notice that the y+1 does not become 4y+4

2.3 More on Solving Linear Equations: Clearing Fractions Multiply both sides by the Least Common Denominator (in this case the LCD = 4):

2.3 More on Solving Linear Equations: Clearing Decimals Multiply both sides by the smallest power of 10 that gets rid of all the decimals

2.3 More on Solving Linear Equations: Why Clear Fractions? It makes the calculation simpler:

2.3 More on Solving Linear Equations 1 – Multiply on both sides to get rid of fractions 2 – Use the distributive property to remove parenthesis 3 – Combine like terms 4 – Put variables on one side, numbers on the other by adding/subtracting on both sides 5 – Get “x” by itself on one side by multiplying both sides 6 – Check your answers

2.3 More on Solving Linear Equations Example:

2.4 An Introduction to Applications for Linear Equations 1 – Decide what you are asked to find 2 – Write down any other pertinent information (use other variables, draw figures or diagrams ) 3 – Translate the problem into an equation. 4 – Solve the equation. 5 – Answer the question posed. 6 – Check the solution.

2.4 An Introduction to Applications for Linear Equations Find the measure of an angle whose complement is 10 larger. x is the degree measure of the angle. 90 – x is the degree measure of its complement 90 – x = 10 + x Subtract 10: 80 – x = x Add x: 80 = 2x Divide by 2: x = 40 The measure of the angle is 40  Check: 90 – 40 = 10 + 40

2.5 Formulas - examples A = lw I = prt P = a + b + c d = rt V = LWH C = 2r Area of rectangle Interest Perimeter of triangle Distance formula Volume – rectangular solid Circumference of circle

2.5 Formulas Example: d=rt; (d = 252, r = 45) then 252 = 45t divide both sides by 45:

2.6 Ratios and Proportions Ratio – quotient of two quantities with the same units Examples: a to b, a:b, or Note: percents are ratios where the second number is always 100:

2.6 Ratios and Proportions Proportion – statement that two ratios are equal Examples: Cross multiplication: if then

2.6 Ratios and Proportions Solve for x: Cross multiplication: so x = 63

2.7 More about Problem Solving Percents are ratios where the second number is always 100: Example: If 70% of the marbles in a bag containing 40 marbles are red, how many of the marbles are red?: #red marbles =

2.7 More about Problem Solving How many gallons of a 12% indicator solution must be mixed with a 20% indicator solution to get 10 gallons of a 14% solution? Let x= #gallons of 12% solution, then 10-x= #gallons of 20% solution :

2.8 Solving Linear Inequalities < means “is less than”  means “is less than or equal to” > means “is greater than”  means “is greater than or equal to” note: the symbol always points to the smaller number

2.8 Solving Linear Inequalities A linear inequality in one variable can be written in the form: ax < b (a0) Addition property of inequality: if a < b then a + c < b + c

2.8 Solving Linear Inequalities Multiplication property of inequality: If c > 0 then a < b and ac < bc are equivalent If c < 0 then a < b and ac > bc are equivalent note: the sign of the inequality is reversed when multiplying both sides by a negative number

2.8 Solving Linear Inequalities Example: -9