Solving Equations. A: Solving One-Variable Equations. One Step x + 6 = 9-3y = 15. Two Step 2a 3 6. Algebra 2 Chapter 1 Notes 1.4 Solving Equations

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1 Algebra 2 Chapter 1 Notes 1.4 Solving Equations 1.4 Solving Equations Topics: Solving Equations Translating Words into Algebra Solving Word Problems A: Solving One-Variable Equations The equations below are easy types of equations that you solved often in Algebra I. One Step x + 6 = 9-3y = 15 Expectations: I want to see for any problem: The original problem Any key steps in getting to your solution- the work Clearly stated solution Answers: Should use original variable if applicable x = 2 or y = 5, etc. FRACTIONS should always be reduced to lowest terms. DECIMALS only if they are terminating and you write the entire thing never round unless the directions say so. Two Step 2a 3 6 In Algebra II, you will face some equations that are bit more challenging to solve. Example 1: Solve each of the following. Show your work! 1a. 6x 5 7 9x 1b v *Get like denominators OR use a scientific calculator with a fraction button* 1

2 Algebra 2 Chapter 1 Notes 1.4 Solving Equations 1c. 5(6 4 y) y 21 1d. 53 = 3(y 2) (3y 1) 1 1f. 2 1e. x 1 x 2 B: Translating Algebra to Words and back (review lesson 1.3) TIPS: A number means the variable pick one! I like x and n, but any letter will do! Try to pick out words that mean operators (+,,, ) is means = Use ( ) to group a sum (etc) together. Example 2: Write an algebraic expression to represent each verbal expression. 2a. three less than a number 2b. six times the cube of a number is the quotient of the same number and 81 2c. the square of a number decreased by the product of the same number and five 2d. twice the difference of a number and six is equal to 24 2

3 Algebra 2 Chapter 1 Notes 1.4 Solving Equations Example 3: Write a verbal expression to represent each equation. 3a. 6 = 5 + x 3b. 7y 2 = 19 m 3c. 3(2m 1) 5 C: Word Problems TIPS: Read the expression out loud that is what you should write down. Write a number for any variable A group in parenthesis is usually a sum difference product or quotient You must SPELL the first word of any sentence even if it is a number. After that, you may use numerals. Procedure for solving word problem 1. Relax. Read through the problem at least twice. 2. Identify a variable. Call it whatever you want, x, n, etc. and tell what it means/stands for (example: let x = the 5 th test score) 3. Write an equation that contains that variable by translating the English words into Math symbols and numbers 4. Solve the equation. 5. Answer the question asked and make sure your answer makes sense Solve each of the following story problems. Be sure to name a variable, show your equation and your work, and answer the question being asked in words. 1. a) The sum of three consecutive integers is 78. What are the integers? b) How would you change the variables for Three consecutive odd integers? 3

4 Algebra 2 Chapter 1 Notes 1.4 Solving Equations 2. Sam s scores on his first four tests are 84, 75, 86, and 92. What must he score on the fifth test to average 88%? 3. Tickets to A Midsummer Night s Dream cost $10.50 for adults and $7.50 for students. Mrs. Smith ordered $192 worth of tickets for a field trip for her English class. If a total of 24 tickets were ordered, how many of each type of ticket did she order? 4. The length of a rectangle is twice as long as the width. Find the dimensions of the rectangle if the perimeter is 360 ft. 5. The sides of a triangle are in a ratio of 4:5:6. The perimeter of the triangle is 52.5 inches. Find the lengths of each side. 4

5 Algebra 2 Chapter 1 Notes 1.5 Solving Inequalities 1.5 Solving Inequalities A: Writing, Solving, and Graphing Inequalities Inequality Summary greater than less than greater than or equal less than or equal Inequalities represent values that may not necessarily be equal. Example: What inequality represents: Fourteen minus 5 times a number is no more than 20? You may recall from Algebra 1, that to graph an inequality in one variable, you use a closed or open dot on the number and then shade to the left or right. Solving inequalities is the same as solving equations EXCEPT if you multiply or divide by a negative number, you have to FLIP the inequality symbol. Solve the example from above, then graph the solution. Examples 1: Solve each inequality. Graph the solution. 1a. 1b. 1c. 5

6 Algebra 2 Chapter 1 Notes 1.5 Solving Inequalities 1d. ** 1e.** B: Solving Compound Inequalities There are two types of compound inequalities: The AND and the OR Type 1: AND Compound Inequalities A B A and B means the overlapping values common to A and B. Example 2: Solve each compound inequality. Graph the solution. 2a. 3x 12 and 8x 16 6

7 Algebra 2 Chapter 1 Notes 1.5 Solving Inequalities 2b. 3x 4 16 and 2x c. 13 2x d. 19 3y e. 5x 15 and x 6 8 Type 2: OR Compound Inequalities A B A or B means the all the area covered by A and all area covered by B 7

8 Algebra 2 Chapter 1 Notes 1.5 Solving Inequalities Example 3: Solve each compound inequality. Graph the solution. 3a. y 2 3 or y 4 3 3b. 4d 1 9 or 2d c. 3x 9 or 2x 10 Example 4: 4a. A farmer wants to make a rectangular pen using no more than 400 ft. of fencing. If he wants the length to be exactly 10 feet longer than the width, describe the possible dimensions of the pen. Write an inequality and solve. 4b. Write an inequality for the following situation: This medicine should be stored between 65 F and 75 F for optimum life. 8

9 Algebra 2 Chapter 1 Notes 1.6 Absolute Value Equations and Inequalities 1.6 Solving Absolute Value Equations and Inequalities A: Solving Absolute Value Equations Perhaps you remember from a previous math class the concept of absolute value. Solve this equation: x 5 Strategy: x A Set up 2 equations: x A or x A Example 1: Solve the following absolute value equation. Be sure to check your answers. The universal symbol for no solution is 1a. x b. a c. 1 3 x d. 5 2 x

10 Algebra 2 Chapter 1 Notes 1.6 Absolute Value Equations and Inequalities x 1f w 7 1e h y 9 1g. x 6 3x 2 B: Solving Absolute Value Inequalities List the possible integer values of x that make x 5 What do you notice? 10

11 Algebra 2 Chapter 1 Notes 1.6 Absolute Value Equations and Inequalities There are two types of absolute value inequalities: Less than and Greater than GO L.A.! can help you remember the difference Greater than = Rewrite and solve like an Or inequality. Less than = Rewrite and solve like an And inequality. Example 2: Solve each inequality. Graph the solution. 2a. 4x b. 3x c. 4s d. 10 2k 2 11

12 Algebra 2 Chapter 1 Notes 1.6 Absolute Value Equations and Inequalities 2e. 3b 5 2 Example 3: Example 4: A chemist is preparing a solution that must contain 3 grams of acid. The acid tolerance for the solution is described by the inequality a What is the weakest acid solution allowed? 12