Equations and Inequalities

10 Key excerpts on "Equations and Inequalities"

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  GRE All the Quant

  Effective Strategies & Practice from 99th Percentile Instructors

  • (Author)
  • 2023(Publication Date)
  • Manhattan Prep

    (Publisher)

  CHAPTER 7 Inequalities and Systems of Equations Now that you’re more familiar with exponents and substitution, it’s time to tackle inequalities and more advanced equations.

  Inequalities behave similarly to equations in most ways, but with a few important distinctions. What is written on one side of the inequality has a relationship to what is on the other side, but the two sides are not defined as strictly equal. For example:

  3x − 5 ≤ 6x − 11

  The left side, 3x − 5, is less than or equal to the right side, 6x − 11. In this chapter, you’ll learn how to manipulate and solve this and more complicated inequalities, as well as how to solve when you are given an inequality alongside an absolute value equation, a regular equation, or a second inequality.

  Interpreting Inequalities

  Inequalities are expressions that use <, >, ≤, or ≥ to describe the relationship between two values. Each of these signs has a distinct meaning:

  <	Less than
  >	Greater than
  ≤	Less than or equal to (or at most)
  ≥	Greater than or equal to (or at least)

  Translating inequalities correctly is essential. Read them from left to right:

  	Read as...	or...
  5 > 4	5 is greater than 4.
  y ≤ 7	y is less than or equal to 7.	y is at most 7.
  x < 5	x is less than 5.
  2x + 3 ≥ 0	2x + 3 is greater than or equal to 0.	2x + 3 is at least 0.

  You can also have two inequalities in one statement (sometimes called a compound inequality):

  	Read as...
  9 < f < 200	9 is less than f, and f is less than 200. Alternatively, f is between 9 and 200.
  −3 < y ≤ 5	−3 is less than y, and y is less than or equal to 5.
  7 ≥ x ≥ 2	7 is greater than or equal to x, and x is greater than or equal to 2. Alternatively, x is between 2 and 7, inclusive.

  Why does the last example use the word inclusive in the alternative reading? And why does the middle example not have an alternative form at all?

  Mathematically, the word inclusive means including. If x is between 2 and 7 inclusive, then 2 and 7 are included in that range; that is, x can be 2 and x can be 7. Contrast that with: f is between 9 and 200. In this case, f must be between those two values but f

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  College Algebra

  • Cynthia Y. Young(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  88 CHAPTER 1 Equations and Inequalities In This Chapter You will solve linear and quadratic equations. You will then solve more complicated equations (polynomial, rational, radical, and absolute value) by first transforming them into linear or quadratic equations. Then you will solve linear, quadratic, polynomial, rational, and absolute value inequalities. Throughout this chapter you will solve applications of Equations and Inequalities. 1.1 Linear Equations SKILLS OBJECTIVES • Solve linear equations in one variable. • Solve rational equations that are reducible to linear equations. CONCEPTUAL OBJECTIVES • Understand the definition of a linear equation in one variable. • Eliminate values that result in a denominator being equal to zero. 1.1.1 Solving Linear Equations in One Variable 1.1.1 Skill Solve linear equations in one variable. 1.1.1 Conceptual Understand the definition of a linear equation in one variable. An algebraic expression (see Chapter 0) consists of one or more terms that are combined through basic operations such as addition, subtraction, multiplication, or division; for example: 3x + 2 5 − 2y x + y An equation is a statement that says two expressions are equal. For example, the following are all equations in one variable, x: x + 7 = 11 x 2 = 9 7 − 3x = 2 − 3x 4x + 7 = x + 2 + 3x + 5 To solve an equation means to find all the values of x that make the equation true. These values are called solutions, or roots, of the equation. The first of these statements shown above, x + 7 = 11, is true when x = 4 and false for any other values of x. We say that x = 4 is the solution to the equation. Sometimes an equation can have more than one solution, as in x 2 = 9. In this case, there are actually two values of x that make this equation true, x = −3 and x = 3. • Solve linear equations. • Solve application problems involving linear equations. • Solve quadratic equations. • Solve rational, polynomial, and radical equations.

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  Elementary Algebra

  • Toby Wagner(Author)
  • 2021(Publication Date)
  • Chemeketa Press

    (Publisher)

  CHAPTER 1

  Solving Linear Equations and Inequalities

  Solving equations is at the heart of algebra. Because of this, it seems fitting to begin our study of elementary algebra by learning how to solve equations algebraically. In Chapter 1 , we will focus on solving linear equations, which are typically less complicated than non-linear equations. We will also learn how to solve linear inequalities. As we progress through the chapter, we will be using our skills to help us solve application problems, which are commonly known as “word problems” or “story problems.”

  In this chapter, you’ll study the following topics: 1.1 Solving Linear Equations 1.2 Solving Multi-Step Equations 1.3 Solving Literal Equations 1.4 Solving Linear Inequalities

  1.1 Solving Linear Equations

  Overview

  Here’s a problem:

  The area of a rectangle is the same as the value found by multiplying the length and the width. A rectangle that is 14 feet long has an area of 91 square feet. What is the width of the rectangle?

  This problem states that two things are the same — the area of the rectangle and the value found by multiplying the length and width. Mathematicians write many sentences like this, though usually with mathematical notation instead of words. In math, equations are used to communicate sameness. Equations are the most common sentences in math.

  When you are finished with this section, you will be able to: ◆  Identify various types of equations ◆  Understand the meaning of solutions and equivalent equations ◆  Use addition and subtraction to solve 1-step equations ◆  Use multiplication and division to solve 1-step equations ◆  Solve application problems involving 1-step equations At the end of this section, we will write and solve an equation to find the width of the rectangle in the problem above. In the meantime, let’s learn more about equations.

  A. Types of Equations

  An

  equation

  is a mathematical sentence that asserts that two things are the same or equal. An equals sign (=) means “is the same as.” It’s important, though, to understand that an equation only asserts

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  Beginning Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  5 , 2 x 2 4 # 8 103. 2 7 , 5 a 3 2 8 # 3 104. 2 3 , 4 x 2 9 # 17 For Exercises 105 through 120, identify each problem as an expression, equation or inequality. Simplify each expression, and solve each inequality or equation. Write the solution set of each inequality using interval notation. 105. 2 x 1 7 . 41 106. 6 c 2 10 , 32 107. 4 x 1 7 2 12 x 108. 2 m 2 1 3 2 4 m 2 7 m 2 109. 5 h 1 12 5 3 h 2 16 110. 2 1 4 n 2 7 2 52 5 n 1 19 111. 3 a 1 4 $ 7 a 2 20 112. 2 5 t 2 14 # 3 t 1 2 113. 2 3 x 1 4 3 5 1 3 x 1 7 114. 4 7 d 2 2 7 5 2 d 1 3 7 115. 2 20 , 4 a 1 2 , 12 116 8 # y 4 1 10 , 1 6 117. 2 3 b 1 5 1 4 9 b 2 2 118. 3 5 x 2 8 1 3 10 x 1 1 2 119. 4 , 2 g 2 # 7 120. 2 8 # x 3 # 4 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. C H A P T E R 2 L i n e a r E q u a t i o n s a n d I n e q u a l i t i e s w i t h O n e V a r i a b l e 218 Chapter Summary ● ● An equation is a statement that two expressions are equal. Equations have equal signs. ● ● Solutions to an equation are the values for the variables that make the equation true. ● ● To check solutions to an equation, use the following steps: 1. Substitute in the given value(s) of the variables. 2. Evaluate both sides of the equation. Simplify using the order-of-operations agreement. If both sides are equal then that (those) value(s) of the variable(s) is a (are) solution(s) of the equation. ● ● The addition property of equality: If the same real number or algebraic expression is added to both sides of an equation, the solution to the equation is not changed.

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  PreStatistics

  • Donald Davis, William Armstrong, Mike McCraith, , Donald Davis, William Armstrong, Mike McCraith(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  55 r.nagy/Shutterstock.com Chapter 2 CHAPTER CONTENTS Section 2.1 Translating English to Algebra: Expressions, Equations, and Inequalities Section 2.2 Order of Operations and Evaluating Numerical Expressions Section 2.3 Basics of Solving Linear Equations Algebraic Expressions Used in Statistics and Basics of Solving Equations 56 CHAPTER 2 • Algebraic Expressions Used in Statistics and Basics of Solving Equations SECTION 2.1 Translating English to Algebra: Expressions, Equations, and Inequalities When most people think of arithmetic and algebra, what comes to mind is a page full of numbers and letters, with very few words or sentences associated with them. Statistics is a bit different. Many applications in statistics are based on situations that are presented in narrative form, with our task being to translate that narrative into mathematics. In this section, we learn how to translate English into algebra. OBJECTIVE 1 Differentiate between an Expression and an Equation What differentiates arithmetic from algebra is that algebra is the part of mathematics in which letters are used to represent numbers that are either unknown or change from time to time. These letters are known as variables . Commonly used variables include x , y , z , n , t , and so on. In statistics, some variables are denoted by lower case Greek letters such as m (pronounced “mu”), s (pronounced “sigma”), l (pronounced “lambda”), and x (pronounced “kai”). If we place a real number in front or behind a variable, we call that number a coefficient . For our purposes, many of the coefficients will be represented by rational numbers. Together, variables and coefficients make up algebraic terms . Terms that are separated with addition and subtraction signs are called algebraic expressions .

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  Algebra & Trig

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 144 Chapter 1 Equations, Inequalities, and Mathematical Modeling GO DIGITAL Rational Inequalities The concepts of key numbers and test intervals can be extended to rational inequalities. Use the fact that the value of a rational expression can change sign at its zeros (the x-values for which its numerator is zero) and at its undefined values (the x-values for which its denominator is zero). These two types of numbers make up the key numbers of a rational inequality. When solving a rational inequality, begin by writing the inequality in general form, that is, with zero on the right side of the inequality. EXAMPLE 5 Solving a Rational Inequality Solve 2x - 7 x - 5 ≤ 3. Then graph the solution set. Solution 2x - 7 x - 5 ≤ 3 Write original inequality. 2x - 7 x - 5 - 3 ≤ 0 Write in general form. 2x - 7 - 3x + 15 x - 5 ≤ 0 Find the LCD and subtract fractions. -x + 8 x - 5 ≤ 0 Simplify. Key numbers: x = 5, x = 8 Zeros and undefined values of rational expression Test intervals: (- ∞ , 5), (5, 8), (8, ∞ ) Test: Is -x + 8 x - 5 ≤ 0? Testing these intervals, as shown in the figure below, the inequality is satisfied on the open intervals (- ∞ , 5) and (8, ∞ ).

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  Intermediate Algebra

  • Mark D. Turner, Charles P. McKeague(Authors)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Chapter Outline 2.1 Linear Equations 2.2 Formulas 2.3 Applications 2.4 Linear Inequalities in One Variable and Interval Notation 2.5 Union, Intersection, and Compound Inequalities 2.6 Equations with Absolute Value 2.7 Inequalities Involving Absolute Value 2 iStockphoto.com © Andresr 67 M artina is an international student who is planning on taking some of her college courses here in the U.S. Because her country uses the Celsius scale, she is not familiar with temperatures measured in degrees Fahrenheit. The formula F = 9 __ 5 C + 32 gives the relationship between the Celsius and Fahrenheit temperature scales. Using this formula, we can construct a table that shows the Fahrenheit values for a variety of temperatures measured in degrees Celsius. In this chapter we will see how Martina could use a linear equation or linear inequality to find the Celsius values for temperatures given in degrees Fahrenheit. Degrees Degrees Celsius Fahrenheit 20 ° 68 ° 25 ° 77 ° 30 ° 86 ° 35 ° 95 ° 40 ° 104 ° Equations and Inequalities in One Variable 68 Success Skills 1. Continue to Set and Keep a Schedule Sometimes I find students do well in Chapter 1 and then become overconfident. They will begin to put in less time with their homework. Don’t do it. Keep to the same schedule. 2. Increase Effectiveness You want to become more and more effective with the time you spend on your homework. Increase those activities that are the most beneficial and decrease those that have not given you the results you want. 3. List Difficult Problems Begin to make lists of problems that give you the most difficulty. These are the problems in which you are repeatedly making mistakes. 4. Begin to Develop Confidence with Word Problems It seems that the main difference between people who are good at working word problems and those who are not is confidence. People with confidence know that no matter how long it takes them, they will eventually be able to solve the problem.

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  Intermediate Algebra

  • Jerome Kaufmann, Karen Schwitters, , , Jerome Kaufmann, Karen Schwitters(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Chapter 2 • Equations, Inequalities, and Problem Solving 96 74. Solve each of the following inequalities. (a) 5 x 2 2 . 5 x 1 3 (b) 3 x 2 4 , 3 x 1 7 (c) 4( x 1 1) , 2(2 x 1 5) (d) 2 2( x 2 1) . 2( x 1 7) (e) 3( x 2 2) , 2 3( x 1 1) (f) 2( x 1 1) 1 3( x 1 2) , 5( x 2 3) When we discussed solving equations that involve fractions, we found that clearing the equa-tion of all fractions is frequently an effective technique. To accomplish this, we multiply both sides of the equation by the least common denominator (LCD) of all the fractions in the equa-tion. This same basic approach also works very well with inequalities that involve fractions, as the next examples demonstrate.  Solve 2 3 x 2 1 2 x . 3 4 . Solution 2 3 x 2 1 2 x . 3 4 12 a 2 3 x 2 1 2 x b . 12 a 3 4 b Multiply both sides by 12 , which is the LCD of 3 , 2 , and 4 EXAMPLE 1 ClassroomExample Solve 1 2 m 1 3 4 m , 3 8 . 64. 3( x 1 2) 1 4 , 2 2 x 1 14 1 x 65. 3( x 2 2) 2 5(2 x 2 1) $ 0 66. 4(2 x 2 1) 2 3(3 x 1 4) $ 0 67. 2 5(3 x 1 4) , 2 2(7 x 2 1) 71. Do the less than and greater than relations possess a symmetric property similar to the symmetric property of equality? Defend your answer. 72. Give a step-by-step description of how you would solve the inequality 2 3 . 5 2 2 x . 73. How would you explain to someone why it is necessary to reverse the inequality symbol when multiplying both sides of an inequality by a negative number? 68. 2 3(2 x 1 1) . 2 2( x 1 4) 69. 2 3( x 1 2) . 2( x 2 6) 70. 2 2( x 2 4) , 5( x 2 1) ThoughtsIntoWords FurtherInvestigations AnswerstotheConceptQuiz 1. False 2. True 3. False 4. True 5. False 6. True 7. False 8. True 9. False 10. True O B J E C T I V E S Solve inequalities involving fractions or decimals Solve inequalities that are compound statements Use inequalities to solve word problems 1 2 3 2.6 MoreonInequalitiesandProblemSolving Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

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  College Algebra

  • James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  478 CHAPTER 5 ■ Systems of Equations and Inequalities Graphing Inequalities (pp. 467–468) To graph an inequality: 1. Graph the equation that corresponds to the inequality. This “boundary curve” divides the coordinate plane into separate regions. 2. Use test points to determine which region(s) satisfy the inequality. 3. Shade the region(s) that satisfy the inequality, and use a solid line for the boundary curve if it satisfies the inequality (  or  ) and a dashed line if it does not (  or  ). Graphing Systems of Inequalities (p. 469) To graph the solution of a system of inequalities (or feasible region determined by the inequalities): 1. Graph all the inequalities on the same coordinate plane. 2. The solution is the intersection of the solutions of all the inequalities, so shade the region that satisfies all the inequalities. 3. Determine the coordinates of the intersection points of all the boundary curves that touch the solution set of the system. These points are the vertices of the solution. 1. (a) What is a system of equations in the variables x , y , and z ? (b) What are the three methods we use to solve a system of equations? 2. Consider the following system of equations: e x  y  3 3 x  y  1 (a) Describe the steps you would use to solve a system by the substitution method. Use the substitution method to solve the given system. (b) Describe the steps you would use to solve a system by the elimination method. Use the elimination method to solve the given system. (c) Describe the steps you would use to solve a system by the graphical method. Use the graph shown below to solve the system. y x+y=3 3x-y=1 x 1 1 0 3. What is a system of linear equations in the variables x , y , and z ? 4. For a system of two linear equations in two variables, (a) How many solutions are possible? (b) What is meant by an inconsistent system? (c) What is meant by a dependent system? 5. What operations can be performed on a linear system to arrive at an equivalent system? 6.

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  Intermediate Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. S E C T I O N 2 . 4 S o l v i n g L i n e a r I n e q u a l i t i e s 199 Solving Compound Inequalities Sometimes quantities involved in inequalities are subject to two constraints. These two constraints on the quantity will most often appear as two inequalities separated by the words and or or. Inequalities of this type are called compound inequalities. DEFINITIONS Compound Inequalities ● ● If a , c, then a , t and t , c is a compound inequality. It can be written more compactly as a , t , c, meaning that t is in between a and c. ● ● An inequality of the form a , t or t . c is a compound inequality. Note: The above definitions still hold if the inequality is replaced with a # or $. For example, the normal range for the body mass index (BMI) for a healthy adult over 20 years should be between 18.5 and 24.9, including the endpoints. Another way to say this is that the BMI should be greater than or equal to 18.5 (first constraint) and less than or equal to 24.9 (second constraint). Mathematically, these two constraints on BMI can be expressed in two ways. Let B represent BMI for a healthy adult over 20. Compound Inequality Written as Two Inequalities Compound Inequality Written as One Inequality 18.5 # B and B # 24.9 18.5 # B # 24.9 Writing a compound inequality involving and as one inequality is a more succinct way to write the same inequality with the two constraints. To read aloud 18.5 # B # 24.9, we say “B is in between 18.5 and 24.9, including the endpoints.” To graph the compound inequality 18.5 # B # 24.9, use a number line. Since both endpoints are included in the inequality, use square brackets.

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