3.1 Solving Linear Equations Part I 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 (Subtraction) Property of Equality:

3.1 Solving Linear Equations Part I Multiplication Property of Equality: 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.

3.1 Solving Linear Equations Part I Example: Solve by getting x by itself on one side of the equation. Subtract 7 from both sides: Divide both sides by 3:

3.2 Solving Linear Equations Part II - Fractions/Decimals As with expressions, you need to combine like terms and use the distributive property in equations. Example:

3.2 Solving Linear Equations Part II - Fractions/Decimals Fractions - Multiply each term on both sides by the Least Common Denominator (in this case the LCD = 4): Multiply by 4: Reduce Fractions: Subtract x: Subtract 5:

3.2 Solving Linear Equations Part II - Fractions/Decimals Decimals - Multiply each term on both sides by the smallest power of 10 that gets rid of all the decimals Multiply by 100: Cancel: Distribute: Subtract 5x: Subtract 50: Divide by 5:

3.2 Solving Linear Equations Part II - Fractions/Decimals Eliminating fractions makes the calculation simpler: Multiply by 94: Cancel: Distribute: Subtract x: Subtract 10:

3.2 Solving Linear Equations Part II 1 – Multiply on both sides to get rid of fractions/decimals 2 – Use the distributive property 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 or dividing on both sides 6 – Check your answers (if you have time)

3.2 Solving Linear Equations Part II Example: Clear fractions: Combine like terms: Get variables on one side: Solve for x:

3.3 Applications of Linear Equations to General Problems 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.

3.3 Applications of Linear Equations to General Problems Example: The sum of 3 consecutive integers is 126. What are the integers? x = first integer, x + 1 = second integer, x + 2 = third integer

3.3 Applications of Linear Equations to General Problems Example: Renting a car for one day costs $20 plus $.25 per mile. How much would it cost to rent the car for one day if 68 miles are driven? $20 = fixed cost, $.25  68 = variable cost

3.4 Percent Increase/Decrease and Investment Problems A number increases from 60 to 81. Find the percent increase in the number.

3.4 Percent Increase/Decrease and Investment Problems A number decreases from 81 to 60. Find the percent increase in the number. Why is this percent different than the last slide?

3.4 Percent Increase/Decrease and Investment Problems A flash drive is on sale for $12 after a 20% discount. What was the original price of the flash drive?

3.4 Percent Increase/Decrease and Investment Problems Another Way: A flash drive is on sale for $12 after a 20% discount. What was the original price of the flash drive? Since $12 was on sale for 20% off, it is 100% - 20% = 80% of the original price set up as a proportion (see 3.6):

3.4 Percent Increase/Decrease and Investment Problems Simple Interest Formula: I = interest P = principal R = rate of interest per year T = time in years

3.4 Percent Increase/Decrease and Investment Problems Example: Given an investment of $9500 invested at 12% interest for 1½ years, find the simple interest.

3.4 Percent Increase/Decrease and Investment Problems Example: If money invested at 10% interest for 2 years yields $84, find the principal.

3.5 Geometry Applications and Solving for a Specific Variable A = lw P = a + b + c Area of rectangle Area of a triangle Perimeter of triangle Sum of angles of a triangle Area of a circle Circumference of a circle

3.5 Geometry Applications and Solving for a Specific Variable Complementary angles – add up to 90 Supplementary angles – add up to 180 Vertical angles – the angles opposite each other are congruent

3.5 Geometry Applications and Solving for a Specific Variable Find the measure of an angle whose complement is 10 larger. x = degree measure of the angle. 90 – x = measure of its complement 90 – x = 10 + x Subtract 10: Add x: Divide by 2:

3.6 Applications of Linear Equations to Proportions, d=rt, and Mixture Problems Ratio – quotient of two quantities with the same units Note: percents are ratios where the second number is always 100:

3.6 Applications of Linear Equations to Proportions, d=rt, and Mixture Problems Percents : Example: If 70% of the marbles in a bag containing 40 marbles are red, how many of the marbles are red?: # of red marbles =

3.6 Applications of Linear Equations to Proportions, d=rt, and Mixture Problems Proportion – statement that two ratios are equal: Solve using cross multiplication:

3.6 Applications of Linear Equations to Proportions, d=rt, and Mixture Problems Solve for x: Solution:

3.6 Applications of Linear Equations to Proportions, d=rt, and Mixture Problems Example: d=rt (distance = rate  time) How long will it take to drive 420 miles at 50 miles per hour?

3.6 Applications of Linear Equations to Proportions, d=rt, and Mixture Problems General form of a mixture problem: x units of an a% solution are mixed with y units of a b% solution to get z units of a c% solution Equations will always be:

3.6 Applications of Linear Equations to Proportions, d=rt, and Mixture Problems Example: How many gallons of a 10% indicator solution must be mixed with a 20% indicator solution to get 10 gallons of a 14% solution? Let x = # gallons of 10% solution, then 10 - x = # gallons of 20% solution :

3.7 Solving Linear Inequalities in One Variable < 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

3.7 Solving Linear Inequalities in One Variable 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

3.7 Solving Linear Inequalities in One Variable 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

3.7 Solving Linear Inequalities in One Variable Example: -9

3.8 Solving Compound Inequalities For any 2 sets A and B, the intersection of A and B is defined as follows: AB = {x  x is an element of A and x is an element of B} For any 2 sets A and B, the union of A and B is defined as follows: AB = {x  x is an element of A or x is an element of B}

3.8 Solving Compound Inequalities Example: 1 3

3.8 Solving Compound Inequalities Example: 1 3