Page 1 : Chapter 3, , CHAPTER 3: LINEAR EQUATIONS AND INEQUALITIES, Chapter Objectives, By the end of this chapter, the student should be able to,  Solve linear equations (simple, dual-side variables, infinitely many solutions or no, solution, rational coefficients),  Solve linear inequalities,  Solve literal equations with several variables for one of the variables, , Contents, CHAPTER 3: LINEAR EQUATIONS AND INEQUALITIES ............................................................................. 107, SECTION 3.1: LINEAR EQUATIONS ........................................................................................................ 108, A., , VERIFYING SOLUTIONS ............................................................................................................. 108, , B., , ONE-STEP EQUATIONS ............................................................................................................. 108, , C., , TWO-STEP EQUATIONS ............................................................................................................ 110, , D., , GENERAL EQUATIONS .............................................................................................................. 111, , E., , SOLVING EQUATIONS WITH FRACTIONS ................................................................................. 113, , EXERCISES ......................................................................................................................................... 115, SECTION 3.2: LINEAR INEQUALITIES ..................................................................................................... 116, A., , GRAPHING LINEAR INEQUALITIES............................................................................................ 116, , B., , SOLVING LINEAR INEQUALITIES ............................................................................................... 117, , C., , TRIPARTITE INEQUALITIES........................................................................................................ 119, , EXERCISES ......................................................................................................................................... 121, SECTION 3.3: LITERAL EQUATIONS ....................................................................................................... 122, A., , SOLVING FOR A VARIABLE ....................................................................................................... 122, , EXERCISES ......................................................................................................................................... 123, CHAPTER REVIEW.................................................................................................................................. 124, , 107

Page 2 : Chapter 3, , SECTION 3.1: LINEAR EQUATIONS, A. VERIFYING SOLUTIONS, A linear equation is made up of two expressions that are equal to each other. A linear equation may have, one or two variables in it, where each variable is raised to the power of 1. No variable in a linear equation, can have a power greater than 1., Linear equation:, Not a linear equation:, , 2𝑦𝑦 = 3𝑥𝑥 + 1, (each variable in the equation is raised to the power of 1), 𝑦𝑦 2 = 3𝑥𝑥 + 1, (y is raised to the power of 2, therefore this is not linear), , The solution to an equation is the value, or values, that make the equation true. Given a solution, we plug, the value(s) into the respective variable(s) and then simplify both sides. The equation is true if both sides, of the equation equal each other., , MEDIA LESSON, , Is it a solution? (Duration 5:00), View the video lesson, take notes and complete the problems below, A solution to an equation is the _______________for the _______________ that makes the equation, _______________. To test a possible solution, _______________ the _______________ with the, _______________., Example. Is 𝑎𝑎 = 3 the solution to 4𝑎𝑎 − 18 = 2𝑎𝑎? Explain your answer., , ___________________________________________________________________________, , YOU TRY, a) Verify that 𝑥𝑥 = −3 is a solution to the algebraic equation 5𝑥𝑥 − 2 = 8𝑥𝑥 + 7., b) Is 𝑚𝑚 = −1 a solution to the algebraic equation 𝑚𝑚 + 9 = 3𝑚𝑚 + 5?, c) Is 𝑎𝑎 = 5 a solution to the algebraic equation −4(𝑎𝑎 + 1) = 6(1 − 𝑎𝑎)?, , B. ONE-STEP EQUATIONS, The Addition Property of Equality, If 𝒂𝒂 = 𝒃𝒃, then for any number 𝒄𝒄,, , 𝒂𝒂 + 𝒄𝒄 = 𝒃𝒃 + 𝒄𝒄, That is, if we are given an equation, then we are allowed to add the same number to both sides of the, equation to get an equivalent statement., , 108

Page 3 : Chapter 3, MEDIA LESSON, Addition Principle (Duration 5:00), View the video lesson, take notes and complete the problems below, To clear a negative we _______________ it to _______________., Example (follow the structure in the video and fill in the diagram below), 𝑥𝑥 − 9 = 4, , The Multiplication Property of Equality, If 𝒂𝒂 = 𝒃𝒃, then for any number 𝒄𝒄,, , 𝒂𝒂 ∙ 𝒄𝒄 = 𝒃𝒃 ∙ 𝒄𝒄, That is, if we are given an equation, then we are allowed to multiply by the same number on both, sides of the equation to get an equivalent statement., We use these two properties to help us solve an equation. To solve an equation means to “undo” all the, operations of the equation, leaving the variable by itself on one side. This is known as isolating the, variable., MEDIA LESSON, Multiplication (Division) Principle (Duration 5:00), View the video lesson, take notes and complete the problems below, To clear multiplication we _______________ both sides by the _______________., Example (follow the structure in the video and fill in the diagram below), −8𝑥𝑥 = 72, , 109

Page 4 : Chapter 3, ,  NOTE: When using the Multiplication Property of Equality on an equation like, , −𝑥𝑥 = 4, It is easier to think of the negative in front of the variable as a −1 being multiplied by 𝑥𝑥, that is, −1 ∙ 𝑥𝑥 = 4, We then multiply both sides by −1 to isolate the variable., (−1) ∙ −1 ∙ 𝑥𝑥 = 4 ∙ (−1), ⟹ 1 ∙ 𝑥𝑥 = −4, ⟹ 𝑥𝑥 = −4, When using the Multiplication Property of Equality on an equation where the coefficient is a number, other than 1, 3𝑥𝑥 = 3, We take the coefficient’s reciprocal then multiply both sides of the equation by that reciprocal. This, will isolate the variable, that is, 1, 1, � � ∙ 3 ∙ 𝑥𝑥 = 3 ∙ � �, 3, 3, 3, ⟹ 1 ∙ 𝑥𝑥 =, 3, ⟹ 𝑥𝑥 = 1, YOU TRY, Solve., a) 𝑥𝑥 + 7 = 18, , b) 𝑟𝑟 − 4 = 5, , c) −4 + 𝑏𝑏 = 45, , d) 3 = 19 + 𝑚𝑚, , e) −3𝑦𝑦 = −42, , f), , −5 = −𝑥𝑥, , C. TWO-STEP EQUATIONS, Steps to solve a linear two-step equation., 1. Apply the Addition Property of Equality., 2. Apply the Multiplication Property of Equality to isolate the variable., 3. Check by substituting your answer into the original equation., , MEDIA LESSON, Basic Two Step (Duration 4:59), View the video lesson, take notes and complete the problems below, 110

Page 5 : Chapter 3, Simplifying we use order of operations and we _______________ before we _______________. Solving, we work in reverse so we will _______________ first and then _______________ second., Example (follow the structure in the video and fill in the diagram below), −9 = −5 − 2𝑥𝑥, , YOU TRY, Solve for the variable in each of the following equations. Check your answers., a) Solve: 2𝑏𝑏 − 4 = 12, , Check:, , b) Solve: 4 + 3𝑟𝑟 = 5, , Check:, , c) Solve: 3 = 19 − 2𝑚𝑚, , Check:, , d) Solve: 11 − 𝑦𝑦 = 32, , Check:, , D. GENERAL EQUATIONS, We will now look at some more general linear equations, that is, equations that require more than two, steps to solve. These equations may have more than one of the same variable on each side of the equal, sign, , and/or may contain parentheses, , 𝑥𝑥 − 5 = 4𝑥𝑥 + 7, 3(4𝑛𝑛 − 2) = 5(𝑛𝑛 + 3), , MEDIA LESSON, General Equations (Duration 5:00), , View the video lesson, take notes and complete the problems below, Move variables to one side by, ______________________________________________________________., 111

Page 6 : Sometimes we may have to _______________ first. Simplify by _______________ and, , Chapter 3, , _______________ on each side., Example (follow the structure in the video and fill in the diagram below), 2𝑥𝑥 + 7 = −5𝑥𝑥 − 3, , Use the following steps to solve a general equation., 1. Simplify each side of the equation. Remove parentheses if necessary. Combine like terms., 2. Add terms on each side of the equation so that all terms containing the variable are on one side of, the equal sign and all constant terms are on the other side., 3. Simplify each side of the equation by combining like terms., 4. Apply the Multiplication Property of Equality to isolate the variable., 5. Check by substituting the solution into the original equation., YOU TRY, Solve for the variable in each of the following equations. Check your answers., a) Solve: 𝑥𝑥 − 5 = 4𝑥𝑥 + 7, , Check:, , b) Solve: 3(4𝑛𝑛 − 2) = 5(𝑛𝑛 + 3), , Check:, , 112

Page 7 : c) Solve: 4 − (2𝑦𝑦 − 1) = 2(5𝑦𝑦 + 9) + 𝑦𝑦, , Check:, , Chapter 3, , E. SOLVING EQUATIONS WITH FRACTIONS, When solving linear equations with fractions, it is vital to remember the Multiplication Property of, Equality. Previously, we’ve only dealt with coefficients that were integers. Now we will be looking at, coefficients that are rational numbers., 5𝑥𝑥, = −5, 6, , We can manipulate the left side of this equation as such, , 5, ∙ 𝑥𝑥 = −5, 6, , Looking at it this way, we can then use the Multiplication Property of Equality and multiply both sides of, the equation by the coefficient’s reciprocal, 6 5, 6, � � ∙ ∙ 𝑥𝑥 = −5 ∙ � �, 5 6, 5, ⟹ 1 ∙ 𝑥𝑥 = −6, ⟹ 𝑥𝑥 = −6, , Another way to solve this type of equation is to clear the fractions in the equation by multiplying by the, LCD., MEDIA LESSON, Distributing with Fractions (Duration 5:00), View the video lesson, take notes and complete the problems below, Important: Always _______________first and _________________________ second., Solve the equation below by multiplying the equation by the LCD., 2, 7, 5, (𝑥𝑥 + 4) = 5 � 𝑥𝑥 − �, 3, 15, 6, , 113

Page 8 : Chapter 3, , YOU TRY, a) Solve:, , 𝑥𝑥, 6, , b) Solve:, , 3, 𝑎𝑎, 4, , Check:, , = −5, , Check:, , =8, , 5, , 6, , c) Solve: 0 = − 4 �𝑥𝑥 − 5�, , Check:, , 114

Page 9 : Chapter 3, , EXERCISES, Solve for the variable in each of the following equations. Reduce, simplify, and check your answers., Show all steps, and box your answer., 1) 8𝑥𝑥 − 2 = 30, 2) 5 − 𝑥𝑥 = 3, 1, 2, , 3) − 𝑥𝑥 − 4 = 8, 4), , 2, 𝑥𝑥, 3, , + 3 = 15, , 5) 4𝑥𝑥 − 8 = −𝑥𝑥 + 7, 6), , 3, 𝑥𝑥, 4, , 1, 2, , 9, 8, , − = 𝑥𝑥 +, , 3, 2, , 7) 6𝑥𝑥 − 4(−2𝑥𝑥 + 8) = 10, , 8) −2(4𝑥𝑥 − 2) = −2(𝑥𝑥 − 8), 9) (2𝑥𝑥 − 7) − (4𝑥𝑥 + 8) = 4(𝑥𝑥 + 6), 10) 2(4𝑥𝑥 + 3) = 8𝑥𝑥 + 1, 11) 5(𝑥𝑥 + 6) − 𝑥𝑥 = 4(𝑥𝑥 + 7) + 2, 3, 4, , 5, 4, , 12) − 𝑚𝑚 =, 8, 3, , 1, 2, , 113, 24, 4, 3, , 2, 3, , 13) − − 𝑥𝑥 = − 𝑥𝑥 − �−, , 13, 𝑥𝑥, 4, , + 1�, , 115

Page 10 : Chapter 3, , SECTION 3.2: LINEAR INEQUALITIES, A. GRAPHING LINEAR INEQUALITIES, An algebraic inequality is a mathematical sentence connecting an expression to a value, variable, or, another expression with an inequality sign. Below is a table of inequalities we will be using, Symbol, <, >, ≤, ≥, ≠, , In Words, less than, greater than, less than or equal to, greater than or equal to, not equal, , Examples, 1 < 2 “1 is less than 2”, 4 > 3 “4 is greater than 3”, 0 ≤ 5 “0 is less than 5”, −1 ≥ −1 “−1 is equal to −1”, 3 ≠ 4 “3 is not equal to 4”, , A solution to an inequality is a value that makes the inequality true. For example, a solution to the, inequality, 𝑥𝑥 < 1, , may be 0 since 0 is indeed less than 1. However, 2 cannot possibly be a solution since 2 is not less than, 1., NOTE: The inequality symbols < and > can be quite easy to interpret, however, the inequalities, symbols ≤ and ≥ on the other hand, can be tricky. For example,, 𝑥𝑥 ≤ 1, is read as “𝑥𝑥 is less than or equal to 1.” The keyword here is the word “or.” The word “or” tells us that, our solution can be less than 1 or equal to 1. So 0 is a solution to this inequality since 0 is less than 1., As it turns out, 1 is also a solution to this inequality. The solution 1 is not less than 1 but it is, equivalent to 1, thus 1 is a solution., Notice that this reasoning does not work with strict inequalities., To graph an inequality, let us look at 𝑥𝑥 < 1. We first draw a number line and mark the number in our, inequality on the line., 1, , We then draw an open circle or closed circle (depending on the inequality symbol) on the number line,, above the number we marked., , 1, , The final step is to draw a line in the direction of the solutions., , 1, , Remember: We use an open circle ∘ with the symbols < and >, and a closed circle ∙ with the symbols ≤, or ≥, 116

Page 11 : Chapter 3, MEDIA LESSON, Interval Notation (Duration 3:04), View the video lesson, take notes and complete the problems below, Interval notation is used to _____________________ a graph with _______________ numbers., Interval notation will always be read _______________ to ______________., (, , ,, , ), , We use _______________ parentheses for less/greater than, and _______________ for less/greater, than or equal to., The symbols −∞ and ∞ will always use _______________ parentheses., , Example, graph the interval (−∞, −1) on the number line below., , YOU TRY, a) Determine whether the number 4 is a solution to the following inequalities., 𝑥𝑥 > 1, 𝑥𝑥 < 1, 𝑥𝑥 ≤ 9, 𝑥𝑥 > 4, , 𝑥𝑥 ≥ 4, , b) Graph the following inequalities in the box below. Write the solution using interval notation., Inequality, , Graph, , Interval Notation, , 𝑥𝑥 > 2, 𝑥𝑥 ≥ 2, , 𝑥𝑥 < −2, 𝑥𝑥 ≤ −2, , B. SOLVING LINEAR INEQUALITIES, A linear inequality has the form, 𝑎𝑎𝑎𝑎 + 𝑏𝑏 < 𝑐𝑐, , where 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 are real numbers. This definition is the same for ≤, ≥, or >., 117

Page 12 : Chapter 3, To solve linear inequalities we use the following properties (in the following properties we use the <, symbol. Keep in mind that these properties work with the other inequality symbols too):, The Addition Property of Inequalities, For real numbers 𝒂𝒂, 𝒃𝒃, and 𝒄𝒄, if 𝒂𝒂 < 𝒃𝒃, then 𝒂𝒂 + 𝒄𝒄 < 𝒃𝒃 + 𝒄𝒄., The Multiplication Property of Inequalities, For real numbers 𝒂𝒂, 𝒃𝒃, and 𝒄𝒄 > 𝟎𝟎, if 𝒂𝒂 < 𝒃𝒃, then 𝒂𝒂 ∙ 𝒄𝒄 < 𝒃𝒃 ∙ 𝒄𝒄, If 𝒄𝒄 < 𝟎𝟎, then 𝒂𝒂 ∙ 𝒄𝒄 > 𝒃𝒃 ∙ 𝒄𝒄, , When we are multiplying or dividing by a negative number, we reverse the sign of the inequality., , Steps to solve a general equation., 1. Simplify each side of the inequality. Remove parentheses if necessary. Collect like terms., 2. Add terms on each side of the inequality so that all terms containing the variable are on one side, and all constant terms are on the other side., 3. Simplify each side of the inequality by combining like terms., 4. Apply the Multiplication Property of Inequalities to isolate the variable., 5. Check by substituting the solution (endpoint and a value from the solution set) into the original, inequality., , MEDIA LESSON, Solving (Duration 5:00), View the video lesson, take notes and complete the problems below, Solving inequalities is just like ______________________________., The only exception is if you _______________ or _______________ by a _______________, you must, _______________., Example. Solve the inequality below using the video as a guide. Write the solution in interval notation., 7 − 5𝑥𝑥 ≤ 17, , YOU TRY, Solve the inequality, check your answer, and graph the solution on a number line. Give the solution in, interval notation., a) 3𝑥𝑥 > 𝑥𝑥 + 6, 118

Page 13 : Chapter 3, , b) 3 − 5𝑎𝑎 ≤ 2(𝑎𝑎 + 5), , c) −5(𝑥𝑥 + 2) ≥ −3(𝑥𝑥 + 4), , C. TRIPARTITE INEQUALITIES, We have been dealing with inequalities where the variable (or expression containing the variable) is on, the left or the right side of a number. We will now look at a special type of inequality called a tripartite, inequality, where the expression containing the variable is between two numbers, for example, −1 < 𝑥𝑥 < 1, , The solution for this inequality in interval notation is (−1, 1). Graphing the solution we get, , -1, Below is another example., , 1, , −1 ≤ 𝑥𝑥 < 1, , The solution for this inequality in interval notation is [−1, 1). Graphing the solution we get, -1, , 1, , MEDIA LESSON, Tripartite (Duration 5:00), View the video lesson, take notes and complete the problems below, A tripartite inequality is a __________ part inequality. We use a tripartite inequality when our variable is, _______________ two numbers, When solving these type of inequalities we will, __________________________________________________., When graphing, we will graph the inequality _______________________ the numbers., Example. Solve in the inequality below. Write the solution in interval notation., 5 < 5 − 4𝑥𝑥 ≤ 13, 119

Page 14 : Chapter 3, , YOU TRY, a) Which of the following values are in the solution set for −3 ≤ 𝑛𝑛 < 5?, , 𝑛𝑛 = −5, 𝑛𝑛 = −3, 𝑛𝑛 = 0, 𝑛𝑛 = 4.9, 𝑛𝑛 = 5, 𝑛𝑛 = 12, b) Write a compound inequality to represent the following situation. Clearly indicate what the variable, represents., A number is greater than or equal to 5 but les than 8, , 120

Page 16 : Chapter 3, , SECTION 3.3: LITERAL EQUATIONS, A. SOLVING FOR A VARIABLE, In this section will be constructing linear equations from sentences. Use the steps below as a guide, when approaching each problem., Steps for Writing and Solving Equations, 1) Read and understand the problem. Underline the givens and circle the goal., 2) Form a strategy to solve the problem., 3) Choose a variable to represent the unknown quantity., 4) Read every word in the problem, and translate the given information into an algebraic equation., 5) Solve the equation., 6) Write your answer in a complete sentence., Example:, The cost of leasing a new Ford mustang is $2,311 for a down payment and processing fee plus $276 per, month. For how many months can you lease this car with $10,000?, Solution:, Step 1. The cost of leasing a new Ford mustang is $2,311 for a down payment and processing fee plus, $276 per month. For how many months can you lease this car with $10,000?, Step 2. Since $2,311 is a down payment, this number must be constant, in other words, this number, does not change no matter how many months go by., The $276 does change as the months go by. In the first month we pay $276 on top of the down, payment. In the second month we pay $276 + $276 plus the down payment, and so on. Take note that, the down payment is a one time payment not a monthly payment like the $276., Step 3. We will let 𝑚𝑚 be our variable to represent the number of months we can lease the car., , Step 4. By the information that is given, the linear equation is 276𝑚𝑚 + 2311 = 10,000., , Step 5. Solving for 𝑚𝑚 we get 27.8 months. So how many months can we lease the car? If we say we can, lease the car for 28 months (rounding up) then we would go over the $10,000 limit. So we must round, down and say 27 months., Step 6. We can lease this car for 27 months., , YOU TRY, a) You have just bought a new Sony 55” 3D television set for $1,600. The value of the television set, decreases by $250 per year. How long before the television set is worth half of its original value?, , 122

Page 17 : Chapter 3, , EXERCISES, For each of the following, underline the Givens and circle the Goal of the problem. Form a Strategy, Solve,, and Check. Show all work, and write your answers in complete sentences., 1) John is a door to door vacuum salesman. His weekly salary, 𝑆𝑆, is $200 plus $50 for each vacuum he, sells. This can be written as 𝑆𝑆 = 200 + 50𝑣𝑣, where 𝑣𝑣 is the number of vacuums sold. If John earns, $1000 for a week’s work, how many vacuums did he sell?, 2) Paul is planning to sell bottled water at the local Lollapalooza. He buys 2 crates of water (2000 bottles), for $360 and plans on selling the bottles for $1.50 each. Paul’s profits, 𝑃𝑃 in dollars, from selling 𝑏𝑏, bottles of water is given by the formula 𝑃𝑃 = 1.5𝑏𝑏 − 360. How many bottles does Paul need to sell in, order to break even?, 3) A new Sony 55” 3D television costs $2,499. You are going to pay $600 as a down payment, and pay, the rest in equal monthly installments for one year. Write an equation to represent this situation,, and use it to determine how much you should pay each month. Clearly indicate what the variable in, your equation represents. Solve the equation, and write your answer in a complete sentence., 4) Your yard is a mess, and you decide to hire a landscaper. The Greenhouse charges a $20 consultation, fee plus $11 per hour for the actual work. Garden Pros does not charge a consulting fee, but charges, $15 per hour for the actual work. Write an equation that will help you determine the number of hours, at which the two companies charge the same. Clearly indicate what the variable represents. Solve, the equation, and write your answer in a complete sentence., 5) Let 𝑝𝑝 represent the marked price of an item at Toys R Us. Emma’s aunt gave her a $50 gift card to Toys, R Us for her birthday. If sales tax is currently 9%, set up an equation to express how much she can, spend using her gift card. Solve the equation, and interpret your answer in a complete sentence., 6) Carlos recently hired a roofer to do some necessary work. On the final bill, Carlos was charged a total, of $1105, where $435 was listed for parts and the rest for labor. If the hourly rate for labor was $67,, how many hours of labor was needed to complete the job?, , 123

Page 18 : Chapter 3, , CHAPTER REVIEW, KEY TERMS AND CONCEPTS, Look for the following terms and concepts as you work through the workbook. In the space below, explain, the meaning of each of these concepts and terms in your own words. Provide examples that are not identical, to those in the text or in the media lesson., Linear Equation, Solution to, a Linear Equation, Addition Property, Of Equality, Multiplication Property of, Equality, Algebraic Inequality, , <, ≤, ≥, >, , Addition Property of, Inequalities, Multiplication Property of, Inequalities, ∘, ∙, , 124