MATH 0030 Lecture Notes Section 2.1 The Addition Property of Equality Section 2.2 The Multiplication Property of Equality

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1 MATH 0030 Lecture Notes Section.1 The Addition Property of Equality Section. The Multiplication Property of Equality Introduction Most, but not all, salaries and prices have soared over the decades. To make it easier to compare, the figure below converts historical prices into today s dollars, with adjustments based on consumer s price index. The data for the average annual salary can be described by the mathematical model S 6n 1000 where S is the average U.S. annual salary, in dollars, n years after If trends shown in the formula continue, when will the average annual salary be $50,000? Round to the nearest year.

2 Solving an equation is the process of finding the or that make the equation statement. Linear equation in one variable x is an equation that can be written in the form ax b c where a, b, and c are real numbers and a is not equal to 0 Circle the linear equations in one variable. 3x 7 9 3x x 15x 45 x 6.8 x 6.8 The Addition Property of Equality The same real number (or algebraic expression) may be added to both sides of an equation without changing the equation s solution. This can be expressed symbolically as follows: If a = b, the a + c = b + c Example: Solve the equations. a. x 5 1 b. z c. 1 3 x 4 Example: Solve by combining like terms before using the addition property. 8y 7 7y Example: Solve by isolating the variable to one side of the equation. 3x6 x 5

3 The Multiplication Property of Equality The same nonzero real number (or algebraic expression) may multiply both sides of an equation without changing the solution. This can be expressed symbolically as follows: If a = b and c 0, then ac = bc. Example: Solve the equations. a. 11y 44 b z c. 3 y 16 d. x 3 Example: Solve using both the addition and multiplication properties. a. 4x 3 7 b. 4y 15 5 c. x15 4x 1

4 MATH Lecture Notes Section.3 Solving Linear Equations Introduction: In Massachussetts, speeding fines are determined by the formula F 10( x65) 50 where F is the cost in dollars, of the fine if a person is caught driving x miles per hour. Use this formula to solve, if a fine comes to $400, how fast was the person driving? Steps to Solving a Linear Equation. 1. Simplify the algebraic expression on each side.. Collect all the variable terms on one side and all the constant terms on the other side. 3. Isolate the variable and solve. 4. Check the proposed solution in the original equation. Example: Solve and check. a. 7x 5 3x 16 x 3 b. 8x( x 6) c. 4(x1) 9 3(x 5)

5 Clearing Equations with Fractions: We begin by multiplying sides of the equation by the of all fractions in the equation. The LCD is the number that all the denominators will divide into. Multiplying term on both sides of the equation by the least common denominator will eliminate the fractions in the equation. Example: Solve by clearing fractions. a. 3 x x 39 b. 5 5 x x Clearing Equations with Decimals: It is not necessary with a calculator, but multiplying a decimal number by a power of, has the effect of moving the decimal place to the right Example: Solve by clearing decimals. a. 0.3 x x 1.1 b. 0.48x 3 0. x 6 Recognizing Inconsistent (No Solution) and Identities (All real Numbers) If you attempt to solve an equation with no solution or one that is true for every real number, you will eliminate the variable. An inconsistent equation with NO SOLUTION results in a FALSE statement, such as 5 An identity that is true for ALL REAL NUMBERS results in a TRUE statement, such as 4 4 Example: Solve. 3x 7 3 x 1 a. b. 3 x 1 9 8x 6 5x

6 MATH Lecture Notes Section.4 Formulas and Percents Solving for a Variable means rewriting the formula so that the variable is on one side of the equation. It does not mean obtaining a numerical value for that variable. Example: Solve the formulas for the indicated variables. a. A lw solve for l b. P l w solve for l c. S P Prt solve for r d. Ax By C solve for y Formula s involving Percent: Example: Write an equation then solve. a. What is 9% of 50? b. 9 is 60% of what? c. 18 is what percent of 50?

7 Percent Increase and Decrease Percents are used for comparing changes, such as increases and decreases in sales, population, prices, and production. If a quantity changes, its percent increase or percent decrease can be determined Percent Increase: The increase is what percent of the original amount? Where A, represents the increase, P represents the percent and B represents the original amount. Percent Decrease: The decrease is what percent of the original amount? Where A, represents the decrease, P represents the percent and B represents the original Example: Write an equation then solve. a. A charity has raised $7500, with a goal of raising $500,000. What percent of the goal has been raised? b. Suppose that the state sales tax rate is 9% and you buy a graphing calculator for $96. How much tax is due? What is the calculator s total cost?

8 MATH Lecture Notes Section.5 An Introduction to Problem Solving Introduction The bar graph show the ten most popular college majors with median, or middlemost, starting salaries for recent college graduates. The median starting salary for a nursing major exceeds that of a business major by $10 thousand. Combined, their median starting salaries are $96 thousand. Determine the median starting salaries of business and nursing majors with bachelor s degrees. Define the variables in words: x = Write and algebraic equation: Solve: Sentence Answer: Strategy for Solving Word problems 1. Read the problem carefully. Define variables by letting x represent one of the unknown quantities.. Write an equation in x that translates, or models the conditions of the problem. 3. Solve the equation and answer the problem s question. 4. Check the solution in the original wording of the problem

9 English Phrase Addition The sum of a number and 7 Five more than a number; a number plus 5 A number increased by 6: 6 added to a number Subtraction A number minus 4 A number decreased by 5 A number subtracted from 8 The difference between a number and 6 The difference between 6 and a number Seven less than a number Seven minus a number Nine fewer than a number Multiplication Five times a number The product of 3 and a number Two-thirds of a number (used with fractions) Seventy-five percent of a number(used with decimals) Thirteen multiplied by a number A number multiplied by 13 Twice a number Division A number divided by 3 The quotient of 7 and a number The quotient of a number and 7 The reciprocal of a number More than one operation The sum of twice a number and 7 Twice the sum of a number and 7 Three times the sum of 1 and twice a number Nine subtracted from 8 times a number Twenty-five percent of the sum of 3 times a number and 14 Seven times a number, increased by 4 Seven times the sum of a number and 4 Algebraic Expression x + 7 x + 5 x + 6 x 4 x 5 8 x x 6 6 x x 7 7 x x 9 5x 3x 3 x 0.75x 13x 13x x x 3 7 x x 7 1 x x + 7 (x+7) 3(1 + x) 8x 9 0.5(3x + 14) 7x + 4 7(x+4)

10 Example: The bar graph shows that average rent and mortgage payments in the United States have increased since 1975, even after taking inflation into account. In 008, rent payments averaged $84 per month. For the period shown, monthly rent payments increased by approximately $7 per year. If this trend continues, how many years after 008 will rent payments average $99? In which year will this occur? Define variables: x = Equation: Solve: Sentence Answer: Example: A rectangular swimming pool is three times as long as it is wide. If the perimeter of the pool is 30 feet, what are the pool s dimensions? Define variables: x = Equation: Solve: Sentence Answer:

11 Example: After a 40% price reduction, an exercise machine sold for $564. What was the exercise machine s price before this reduction? Define variables: x = Equation: Solve: Sentence Answer:

12 MATH Lecture Notes Section.6 Problem Solving in Geometry Geometric Formulas for Perimeter and Area Square Area: Square Perimeter: A s P 4s where s is side where s is side Rectangle Area: where l is length and w is width Rectangle Perimeter: P l w where l is length and w is width Triangle Area: 1 A bh where b is base and h is height Trapezoid Area: A 1 ha b where b is base and h is height A lw Geometric Formulas for Circumference and Area of a Circle Area of Circle: A r where r is radius Circumference of Circle: C r where r is radius Geometric Formulas for Volume Cube Volume: 3 V s where s is side Rectangular Solid Volume: V lwh where l is length, w is width, and h is height Circular Cylinder Volume: V r h where r is radius and h is height Sphere Volume: 4 3 V r where r is radius 3 Cone Volume: 1 V r h where r is radius and h is height 3 C r

13 Example: A sailboat has a triangular sail with an area of 4 square feet and a base that is 4 feet long. Find the height of the sail. Example: The diameter of a circular landing pad for helicopters is 40 feet. Find the area and circumference of the landing pad. Express answers in terms of. Then round answers to the nearest square foot and foot respectively. Example: Which one of the following is the better buy: a large pizza with a 16-inch diameter for $15.00 or a medium pizza with an 8-inch diameter for $7.50? Example: A cylinder with a radius of 3 inches and a height of 5 inches has its height doubled. How many times greater is the volume of the larger cylinder than the volume of the smaller cylinder?

14 Angles of a Triangle: The sum of the measures of the three angles of any triangle is. Example: In a triangle, the measure of the first angle is three times the measure of the second angle. The measure of the third angle is 0 degrees less than the second angle. What is the measure of each angle? Complementary and Supplementary Angles Complementary Angles are two angles having a sum of. Supplementary Angles are two angles having a sum of. Algebraic Expressions for Complements and Supplements Measure of an angle: x Measure of the angle s complement: 90-x Measure of the angle s supplement: 180-x Example: The measure of an angle is twice the measure of its complement. What is the angle s measure?

15 MATH Lecture Notes Section.7 Solving Linear Inequalities Graphing solutions to linear inequalities are shown on a number line by shading all points representing numbers that are solutions. SQUARE BRACKETS, [ ], indicate endpoints that solutions. PARENTHESES, ( ), indicate endpoints that solutions. English Sentence x is more than a Inequality Interval Notation x a a, Set-Builder Notation x x a Graph (number lines) x is at least a x a a, x x a x is less than a x a, a x x a x is at most a x a, a x x a Solving a Linear Inequality 1. Simplify the algebraic expression on each side.. Use the addition property of inequality to collect all the variable terms on one side and all the constant terms on the other side. 3. Use the multiplication property of inequality to isolate the variable and solve. Change the of inequality when multiplying or dividing both sides by a number. 4. Express the solution set in interval or set-builder notation, and graph the solution set on a number line. Example: Solve and graph the solution set on a number line. a. 8x 7x 4 b. 6x 18 c. 5y 3 17

16 Example: Solve and graph the solution set on a number line. b. x x a. 6 3x 5x Recognizing Inequalities with No Solution or True for all Real Numbers If you attempt to solve an inequality with no solution or one that is true for every real number, you will eliminate the variable. An inequality with no solution results in a statement, such as 0 >1. The solution set is, the empty set. An inequality that is true for every real number results in a statement, such as 0<1. The solution set is, or x x is a real number. Example: Solve a. 4 x 4x 15 b. 3 x 1 x 1 x Example: You can spend at most $1600 to have a party catered. The caterer charges $95 setup fee and $35 per person. How many people can you invite while staying within your budget?