Chapter 1 Section 5: Equations and Inequalities involving Absolute Value

Size: px
Start display at page:

Download "Chapter 1 Section 5: Equations and Inequalities involving Absolute Value"

Transcription

1 Introduction The concept of absolute value is very strongly connected to the concept of distance. The absolute value of a number is that number s distance from 0 on the number line. Since distance is always positive (or zero), the absolute value is always positive (or zero). Seeing absolute values as distances will make working with them much easier and more intuitive. Note: Watch the animation in the course online to review the concept of absolute value. Example A Solve the equation x 3. Think of the equation as asking What points x on the number line are at a distance of three from zero? Look at the number line: You can see that both 3 and 3 are the required distance from zero. These are the two solutions 3, 3. to the equation above, so the solution set for the equation is { } Solving A B (with B > 0 ) In general, whenever you have an equation of the form A B, with B > 0, then the solutions can be found by solving the following equations: A B and A B. (Note that here A and B represent any algebraic expressions.) The above is true only if B > 0. If B 0, then the only solutions occur when A 0. If B < 0, then there can be no solution because in that case, the left side of the equal sign is positive while the right side is negative (see Example B). No negative number equals any positive number! Example B Solve: x 2. x 2 + Recall that the absolute value of x is a distance, and so it is positive (or zero). And, of course, a positive number can never equal a negative number. So the solution set for this equation is the empty set,. Page 1 of 15

2 Example C Solve: 6x 9 3. Since 3 > 0, the two solutions to this equation can be found by solving the equations: 6x 9 3 and 6x 9 3. Solve the first equation by adding 9 to both sides and then dividing by 6: 6x x 12 6 x x 2 The second equation is also solved by adding 9 to both sides and then dividing by 6: 6x x 6 6 x x 1 The solution set is the set containing the two solutions found: { 1, 2 }. Extended Example 1a Solve: 2x Find the solution set of this equation. Hint: Solve the equation 2x x x 2 2 x x 1 Hint: Solve the equation 2x x x 12 2 x x 6 Hint: The solution set of the equation is the set containing the two solutions you ve just found. 6, 1. The solution set is { } Page 2 of 15

3 Extended Example 1b Solve: 5x 1 9. Find the solution set of this equation. Hint: Can the non-negative quantity on the left side of the equal sign possibly equal the negative quantity on the right side? Since no positive number equals 9, this equation has no solution. The solution set is the empty set,. Extended Example 1c Solve: 7 8. Find the solution set of this equation. Hint: Solve the equation x x 3 Hint: Solve the equation x x 5 Hint: The solution set of the equation is the set containing the two solutions you ve just found. 1 The solution set is,5 3. Page 3 of 15

4 Extended Example 1d Solve: 8x Find the solution set of this equation. Hint: Solve the equation 8x x x 18 8 x x 4 Hint: Solve the equation 8x x x 4 8 x x 2 Hint: The solution set of the equation is the set containing the two solutions you ve just found. 1 9 The solution set is, 2 4. Extended Example 1e Solve: y 1. Find the solution set of this equation. Hint: Can the non-negative quantity on the left side of the equal sign possibly equal the negative quantity on the right side? Since no number greater than or equal to zero can possibly equal 1, this equation can have no solution. The solution set is the empty set,. Page 4 of 15

5 Solving A B The solutions to an equation of the form A B can be found by solving the equations: A B and A B. If B is the opposite of A, then the absolute value of B will equal the absolute value of A (since the absolute value will get rid of the negative). Once again, it helps to think of this absolute value equation in terms of distance. The equation says that the distance from A to 0 is the same as the distance from B to 0. So, either A and B are the same point, or they are opposites on the number line (see Example D). Example D Solve: 5x 9 3 7x. Solutions of this equation can be found by solving equations 5x 9 3 7x and 5x x. ( ) To solve the first equation, first add 7x to both sides and add 9 to both sides: 5x 9 3 7x + 7x x 12x 12 Then, divide both sides by 12: 12 x x 1 To solve the second one, first distribute the negative sign on the right side: ( x) 5x x x. Then subtract 5x and add 3 to both sides: 5x x 5x x 6 2x Divide both sides by 2: 6 2 x x So, the solution set of our absolute value equation is { 3, 1}. Extended Example 2a Solve: 8x 1 8. Find the solution set of this equation. Hint: Solve the equation 8x 1 8, by first adding and 1 to both sides. Page 5 of 15

6 Extended Example 2a, 8x x 9 11 x x 11 Hint: Solve the equation 8x 1 ( 8 ), by first distributing the negative to eliminate parentheses, and then subtracting while adding 1to both sides. 8x 1 8 ( ) 8x x 7 5 x x 5 Hint: The solution set of the equation is the set containing the two solutions you ve just found. 7 9 The solution set is, Extended Example 2b Solve: 5y + 8 9y 2. Find the solution set of this equation. Hint: Solve the equation 5y + 8 9y 2, by first subtracting 5y and adding 2 to both sides. 5y + 8 9y 2 5y + 2 5y y y y Page 6 of 15

7 Extended Example 2b, +, by first distributing the negative to eliminate parentheses, and then adding 9y while subtracting 8 from both sides. 5y + 8 ( 9y 2) 5y + 8 9y y 8 + 9y 8 14y 6 14 y y 7 Hint: Solve the equation 5y 8 ( 9y 2) Hint: The solution set of the equation is the set containing the two solutions you ve just found. 3 5 The solution set is, 7 2. Extended Example 2c Solve: y 12 4y 1. Find the solution set of this equation. Hint: Solve the equation y 12 4y 1, by first subtracting y while adding 1to both sides. y 12 4y 1 y + 1 y y 11 3 y y 3, by first distributing the negative to eliminate parentheses, and then adding 4y and 12to both sides. y 12 ( 4y 1) y 12 4y y y y 13 5 y Hint: Solve the equation y 12 ( 4y 1) y 5 Page 7 of 15

8 Extended Example 2c, Hint: The solution set of the equation is the set containing the two solutions you ve just found The solution set is, 3 5. Solving A < B (with B > 0 ) To solve an inequality of the form A < B, with B > 0, just solve the equivalent three-way inequality: B < A< B. The original inequality says that the distance from A to 0 is less than B. A graph will help you visualize this: So the solution set is the interval ( B B) negative number (or 0 ) is greater than any positive number.,. Note that if B 0, then there can be no solution. No Example E Solve: x < 2. This inequality has no solution, since the left side is positive (or zero), while the right side is negative: x < 2 never + < The solution set for this equation is thus the empty set,. Solving A B (with B 0 ) To solve an inequality of the form A B, with B 0, just solve the equivalent three-way inequality: B A B. In this case, B can be zero, as long as A is also zero. Page 8 of 15

9 Extended Example 3a Solve: 8x Express your solution in set-builder notation and in interval notation. Hint: Write this as a 3-way inequality. 23 8x Hint: Subtract 5 from all three sides. 23 8x x 18 Hint: Divide all three sides by 8. Step 3: 28 8x x x 2 4 Hint: Write the solution in set-builder and in interval notation x x, Extended Example 3b Solve: 6x 7 < 8. Find the solution set of this equation. Hint: Can the non-negative quantity on the left side of the equal sign possibly be less than the negative quantity on the right side? Since no number greater than or equal to zero can possibly be less than 8, this equation can have no solution. The solution set is the empty set,. Extended Example 3c Solve: 7x 6 < 15. Express your solution in set-builder notation and in interval notation. Hint: Write this as a 3-way inequality. 15 < 7x 6 < 15 Hint: Add 6 to all three sides. Page 9 of 15

10 Extended Example 3c, 15 < 7x 6 < < 7x < 21 Hint: Divide all three sides by 7. Step 3: 9 < 7x < x 21 < < x 7 3 < < < x < 3 7 Hint: Write the solution in set-builder and in interval notation. 9 9 x < x < 3, Extended Example 3d Solve: 10 11x < 21. Express your solution in set-builder notation and in interval notation. Hint: Write this as a 3-way inequality. 21 < 10 11x < 21 Hint: Subtract 10 from all three sides. 21 < 10 11x < < 11x < 11 Hint: Divide all three sides by 11. Remember to reverse the inequalities, since we re dividing by a negative. Step 3: 31 11x 11 > > x > > > x > 1 11 Page 10 of 15

11 Extended Example 3d, Hint: Write the solution in set-builder and in interval notation x 1< x < 1, Extended Example 3e Solve: 5x Find the solution set of this equation. Hint: Write this as a 3-way inequality. 0 5x x Hint: The only way this can occur is when 5x + 3 0, because no number can be both greater than zero and less than zero. Solve for x. 5x x 3 5 x x 5 Hint: Write down the solution set of the inequality, which has but one element. 3 5 Solving A > B (with B > 0 ) To solve an inequality of the form A inequalities: A > B or A < B. > B, with B > 0, take the union of the solutions to the two The original inequality is true whenever the distance from A to 0 is greater than B. A graph will help you visualize this: For example, the inequality x > 5 is true whenever the distance from x to 0 is larger than 5. This can happen if x is larger than 5, but can also happen if x is less than 5. Page 11 of 15

12 To see why, imagine for example that x 6. Then, indeed, 6 6 > 5. In general, the inequality A > B has solutions if A is large enough or far enough from 0 in either the positive or negative direction. Note that if B < 0, then every real value of A is a solution. Every positive number (or zero) is larger than any negative number. Example F Solve x > 2. All real numbers are solutions to this inequality: The solution set is thus the set of all real numbers, expressed in interval notation as (, ), and in set-builder notation as { x x } + x > 2 always < <. > Extended Example 4a Solve: Graph your solution, and express it in set-builder notation and in interval notation. Hint: Note the two inequalities without absolute values that will yield the desired solution set or Hint: Solve the first inequality. First, add 10 to both sides. Then, divide both sides by x x 3 Hint: First, graph this solution and then write it in set-builder notation and in interval notation. Step 3: x x, 3 3 Hint: Solve the second inequality. Add 10 to both sides and then divide both sides by 3. Page 12 of 15

13 Extended Example 4a, Step 4: Step 5: 6 3 x x 2 Hint: Graph and then write this solution in set-builder notation and in interval notation. { x x 2 } (, 2] Step 6: Hint: Graph the union of the two solution sets. Hint: Express the solution you graphed in interval and set-builder notation. 14 (, 2 ], { x x 2} x x x x 2 or x 3 3 Extended Example 4b Solve: 2x + 13 > 4. Find the solution set of this equation. Hint: All non-negative quantities are greater than any negative number. No matter what x is, the absolute value is greater than or equal to 0 and thus greater than 4.The,, and in set solution set is thus the set of all real numbers, expressed in interval notation as ( ) builder notation as { x x } < <. Extended Example 4c Solve: 7 2x 1. Graph your solution, and express it in set-builder notation and in interval notation. Hint: Note the two inequalities without absolute values that will yield the desired solution set. 7 2x 1 or 7 2x 1 Hint: Solve the first inequality. First, subtract 7 from both sides. Then, divide both sides by 2. Page 13 of 15

14 Extended Example 4c, 7 2x x 6 2x x 2 3 x 3 Step 3: Hint: First, graph this solution and then write it in set-builder notation and in interval notation. { x x 3 } (,3] Hint: Solve the second inequality, 7 2x 1. Subtract 7 from both sides, and then divide both sides by 2. Step 4: 7 2x 1 7 2x 2x 2 2 x 2 x Step 5: Hint: Graph and then write this solution in set-builder notation and in interval notation. { x x 4} [ 4, ) Step 6: Hint: Graph the union of the two solution sets. Hint: Express the solution you graphed in interval and set-builder notation.,3 4, ( ] [ ) { x x 3} { x x 4} { x x 3 or x 4} Page 14 of 15

15 Extended Example 4d Solve: 4x + 7 > 11. Graph your solution, and express it in set-builder notation and in interval notation. Hint: Note the two inequalities without absolute values that will yield the desired solution set. 4x + 7 > 11 or 4x + 7 < 11 Hint: Solve the first inequality. First, subtract 7 from both sides and then divide both sides by 4. 4x + 7 > x > 4 4 x 4 > 4 4 x > 1 Step 3: Hint: First, graph this solution and then write it in set-builder notation and in interval notation. { x x > 1} ( 1, ) Hint: Solve the second inequality, 4x + 7 < 11. Subtract 7 from both sides and then divide both sides by 4. Step 4: 4x + 7 < x < 18 4 x 2 9 < x < Step 5: 9 2 Hint: Graph and then write this solution in set-builder notation and in interval notation. { x x < 9 } (, 9 ) 2 2 Hint: Graph the union of the two solution sets. Step 6: Hint: Express the solution you graphed in interval and set-builder notation. (, 9 ) ( 1, ) 2 { x x < 9 } { x x > 1} { x x < 9 or x > } End of Lesson Page 15 of 15

Chapter 1 Section 4: Compound Linear Inequalities

Chapter 1 Section 4: Compound Linear Inequalities Chapter 1 Section 4: Compound Linear Inequalities Introduction Compound linear inequalities involve finding the union or intersection of solution sets of two or more linear inequalities. You ve already

More information

Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities . Absolute Value Equations and Inequalities. OBJECTIVES 1. Solve an absolute value equation in one variable. Solve an absolute value inequality in one variable NOTE Technically we mean the distance between

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to, LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are

More information

Inequalities - Absolute Value Inequalities

Inequalities - Absolute Value Inequalities 3.3 Inequalities - Absolute Value Inequalities Objective: Solve, graph and give interval notation for the solution to inequalities with absolute values. When an inequality has an absolute value we will

More information

Section 1. Inequalities -5-4 -3-2 -1 0 1 2 3 4 5

Section 1. Inequalities -5-4 -3-2 -1 0 1 2 3 4 5 Worksheet 2.4 Introduction to Inequalities Section 1 Inequalities The sign < stands for less than. It was introduced so that we could write in shorthand things like 3 is less than 5. This becomes 3 < 5.

More information

1.4 Compound Inequalities

1.4 Compound Inequalities Section 1.4 Compound Inequalities 53 1.4 Compound Inequalities This section discusses a technique that is used to solve compound inequalities, which is a phrase that usually refers to a pair of inequalities

More information

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Method To Solve Linear, Polynomial, or Absolute Value Inequalities: Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

More information

Section 1.1 Real Numbers

Section 1.1 Real Numbers . Natural numbers (N):. Integer numbers (Z): Section. Real Numbers Types of Real Numbers,, 3, 4,,... 0, ±, ±, ±3, ±4, ±,... REMARK: Any natural number is an integer number, but not any integer number is

More information

7. Solving Linear Inequalities and Compound Inequalities

7. Solving Linear Inequalities and Compound Inequalities 7. Solving Linear Inequalities and Compound Inequalities Steps for solving linear inequalities are very similar to the steps for solving linear equations. The big differences are multiplying and dividing

More information

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers. 1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

More information

5.1 Radical Notation and Rational Exponents

5.1 Radical Notation and Rational Exponents Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

More information

2.6 Exponents and Order of Operations

2.6 Exponents and Order of Operations 2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated

More information

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

23. RATIONAL EXPONENTS

23. RATIONAL EXPONENTS 23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,

More information

Algebra 2 Notes AII.7 Functions: Review, Domain/Range. Function: Domain: Range:

Algebra 2 Notes AII.7 Functions: Review, Domain/Range. Function: Domain: Range: Name: Date: Block: Functions: Review What is a.? Relation: Function: Domain: Range: Draw a graph of a : a) relation that is a function b) relation that is NOT a function Function Notation f(x): Names the

More information

Solving Rational Equations and Inequalities

Solving Rational Equations and Inequalities 8-5 Solving Rational Equations and Inequalities TEKS 2A.10.D Rational functions: determine the solutions of rational equations using graphs, tables, and algebraic methods. Objective Solve rational equations

More information

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

More information

Solving systems by elimination

Solving systems by elimination December 1, 2008 Solving systems by elimination page 1 Solving systems by elimination Here is another method for solving a system of two equations. Sometimes this method is easier than either the graphing

More information

Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities Key Concepts: Compound Inequalities Absolute Value Equations and Inequalities Intersections and unions Suppose that A and B are two sets of numbers. The intersection of A and B is the set of all numbers

More information

Practice with Proofs

Practice with Proofs Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

More information

Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010

Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010 Section 2.1: Linear Equations Definition of equation An equation is a statement that equates two algebraic expressions. Solving an equation involving a variable means finding all values of the variable

More information

Determine If An Equation Represents a Function

Determine If An Equation Represents a Function Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The

More information

1.6 The Order of Operations

1.6 The Order of Operations 1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

More information

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

More information

Lesson 7 Z-Scores and Probability

Lesson 7 Z-Scores and Probability Lesson 7 Z-Scores and Probability Outline Introduction Areas Under the Normal Curve Using the Z-table Converting Z-score to area -area less than z/area greater than z/area between two z-values Converting

More information

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

More information

https://williamshartunionca.springboardonline.org/ebook/book/27e8f1b87a1c4555a1212b...

https://williamshartunionca.springboardonline.org/ebook/book/27e8f1b87a1c4555a1212b... of 19 9/2/2014 12:09 PM Answers Teacher Copy Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B Example C #2 Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Students

More information

CAHSEE on Target UC Davis, School and University Partnerships

CAHSEE on Target UC Davis, School and University Partnerships UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

More information

Objective. Materials. TI-73 Calculator

Objective. Materials. TI-73 Calculator 0. Objective To explore subtraction of integers using a number line. Activity 2 To develop strategies for subtracting integers. Materials TI-73 Calculator Integer Subtraction What s the Difference? Teacher

More information

0 0 such that f x L whenever x a

0 0 such that f x L whenever x a Epsilon-Delta Definition of the Limit Few statements in elementary mathematics appear as cryptic as the one defining the limit of a function f() at the point = a, 0 0 such that f L whenever a Translation:

More information

1.2 Linear Equations and Rational Equations

1.2 Linear Equations and Rational Equations Linear Equations and Rational Equations Section Notes Page In this section, you will learn how to solve various linear and rational equations A linear equation will have an variable raised to a power of

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

Adding and Subtracting Positive and Negative Numbers

Adding and Subtracting Positive and Negative Numbers Adding and Subtracting Positive and Negative Numbers Absolute Value For any real number, the distance from zero on the number line is the absolute value of the number. The absolute value of any real number

More information

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

More information

Binary Adders: Half Adders and Full Adders

Binary Adders: Half Adders and Full Adders Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order

More information

Linear Equations and Inequalities

Linear Equations and Inequalities Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................

More information

SIMPLIFYING ALGEBRAIC FRACTIONS

SIMPLIFYING ALGEBRAIC FRACTIONS Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is

More information

Solving Linear Equations - General Equations

Solving Linear Equations - General Equations 1.3 Solving Linear Equations - General Equations Objective: Solve general linear equations with variables on both sides. Often as we are solving linear equations we will need to do some work to set them

More information

The Graphical Method: An Example

The Graphical Method: An Example The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

More information

Chapter 3 Section 6 Lesson Polynomials

Chapter 3 Section 6 Lesson Polynomials Chapter Section 6 Lesson Polynomials Introduction This lesson introduces polynomials and like terms. As we learned earlier, a monomial is a constant, a variable, or the product of constants and variables.

More information

1.7 Graphs of Functions

1.7 Graphs of Functions 64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

Chapter 7 - Roots, Radicals, and Complex Numbers

Chapter 7 - Roots, Radicals, and Complex Numbers Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

More information

5.4 Solving Percent Problems Using the Percent Equation

5.4 Solving Percent Problems Using the Percent Equation 5. Solving Percent Problems Using the Percent Equation In this section we will develop and use a more algebraic equation approach to solving percent equations. Recall the percent proportion from the last

More information

SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property

SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property 498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1

More information

NPV Versus IRR. W.L. Silber -1000 0 0 +300 +600 +900. We know that if the cost of capital is 18 percent we reject the project because the NPV

NPV Versus IRR. W.L. Silber -1000 0 0 +300 +600 +900. We know that if the cost of capital is 18 percent we reject the project because the NPV NPV Versus IRR W.L. Silber I. Our favorite project A has the following cash flows: -1 + +6 +9 1 2 We know that if the cost of capital is 18 percent we reject the project because the net present value is

More information

Session 7 Fractions and Decimals

Session 7 Fractions and Decimals Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

More information

Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

More information

A positive exponent means repeated multiplication. A negative exponent means the opposite of repeated multiplication, which is repeated

A positive exponent means repeated multiplication. A negative exponent means the opposite of repeated multiplication, which is repeated Eponents Dealing with positive and negative eponents and simplifying epressions dealing with them is simply a matter of remembering what the definition of an eponent is. division. A positive eponent means

More information

YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!

YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR! DETAILED SOLUTIONS AND CONCEPTS - DECIMALS AND WHOLE NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST

More information

Linear functions Increasing Linear Functions. Decreasing Linear Functions

Linear functions Increasing Linear Functions. Decreasing Linear Functions 3.5 Increasing, Decreasing, Max, and Min So far we have been describing graphs using quantitative information. That s just a fancy way to say that we ve been using numbers. Specifically, we have described

More information

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have 8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

More information

26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order 26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

More information

3.1 Solving Systems Using Tables and Graphs

3.1 Solving Systems Using Tables and Graphs Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

More information

Pre-Algebra - Integers

Pre-Algebra - Integers 0.1 Pre-Algebra - Integers Objective: Add, Subtract, Multiply and Divide Positive and Negative Numbers. The ability to work comfortably with negative numbers is essential to success in algebra. For this

More information

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

hp calculators HP 17bII+ Net Present Value and Internal Rate of Return Cash Flow Zero A Series of Cash Flows What Net Present Value Is

hp calculators HP 17bII+ Net Present Value and Internal Rate of Return Cash Flow Zero A Series of Cash Flows What Net Present Value Is HP 17bII+ Net Present Value and Internal Rate of Return Cash Flow Zero A Series of Cash Flows What Net Present Value Is Present Value and Net Present Value Getting the Present Value And Now For the Internal

More information

Inequalities - Solve and Graph Inequalities

Inequalities - Solve and Graph Inequalities 3.1 Inequalities - Solve and Graph Inequalities Objective: Solve, graph, and give interval notation for the solution to linear inequalities. When we have an equation such as x = 4 we have a specific value

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

Exponents and Radicals

Exponents and Radicals Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

More information

5 Systems of Equations

5 Systems of Equations Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

More information

The Dot and Cross Products

The Dot and Cross Products The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and

More information

Pre-Algebra - Order of Operations

Pre-Algebra - Order of Operations 0.3 Pre-Algebra - Order of Operations Objective: Evaluate expressions using the order of operations, including the use of absolute value. When simplifying expressions it is important that we simplify them

More information

Activity 1: Using base ten blocks to model operations on decimals

Activity 1: Using base ten blocks to model operations on decimals Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

More information

Domain of a Composition

Domain of a Composition Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such

More information

Let s explore the content and skills assessed by Heart of Algebra questions.

Let s explore the content and skills assessed by Heart of Algebra questions. Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,

More information

1 Solving LPs: The Simplex Algorithm of George Dantzig

1 Solving LPs: The Simplex Algorithm of George Dantzig Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

More information

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

More information

Normal distribution. ) 2 /2σ. 2π σ

Normal distribution. ) 2 /2σ. 2π σ Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

More information

Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions.

Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions. Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions. Basic Functions In several sections you will be applying shifts

More information

Common sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility.

Common sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility. Lecture 6: Income and Substitution E ects c 2009 Je rey A. Miron Outline 1. Introduction 2. The Substitution E ect 3. The Income E ect 4. The Sign of the Substitution E ect 5. The Total Change in Demand

More information

Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20

Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20 SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed

More information

Solving Rational Equations

Solving Rational Equations Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

More information

LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL

LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables

More information

7 Literal Equations and

7 Literal Equations and CHAPTER 7 Literal Equations and Inequalities Chapter Outline 7.1 LITERAL EQUATIONS 7.2 INEQUALITIES 7.3 INEQUALITIES USING MULTIPLICATION AND DIVISION 7.4 MULTI-STEP INEQUALITIES 113 7.1. Literal Equations

More information

6 3 The Standard Normal Distribution

6 3 The Standard Normal Distribution 290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since

More information

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

More information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

Graphing Parabolas With Microsoft Excel

Graphing Parabolas With Microsoft Excel Graphing Parabolas With Microsoft Excel Mr. Clausen Algebra 2 California State Standard for Algebra 2 #10.0: Students graph quadratic functions and determine the maxima, minima, and zeros of the function.

More information

Accentuate the Negative: Homework Examples from ACE

Accentuate the Negative: Homework Examples from ACE Accentuate the Negative: Homework Examples from ACE Investigation 1: Extending the Number System, ACE #6, 7, 12-15, 47, 49-52 Investigation 2: Adding and Subtracting Rational Numbers, ACE 18-22, 38(a),

More information

MATH 90 CHAPTER 1 Name:.

MATH 90 CHAPTER 1 Name:. MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.

More information

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers

More information

Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern.

Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern. INTEGERS Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern. Like all number sets, integers were invented to describe

More information

Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

More information

DIVISIBILITY AND GREATEST COMMON DIVISORS

DIVISIBILITY AND GREATEST COMMON DIVISORS DIVISIBILITY AND GREATEST COMMON DIVISORS KEITH CONRAD 1 Introduction We will begin with a review of divisibility among integers, mostly to set some notation and to indicate its properties Then we will

More information

Week 13 Trigonometric Form of Complex Numbers

Week 13 Trigonometric Form of Complex Numbers Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

More information

More Equations and Inequalities

More Equations and Inequalities Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

More information

Core Maths C1. Revision Notes

Core Maths C1. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

More information

Balancing Chemical Equations

Balancing Chemical Equations Balancing Chemical Equations A mathematical equation is simply a sentence that states that two expressions are equal. One or both of the expressions will contain a variable whose value must be determined

More information

Integers (pages 294 298)

Integers (pages 294 298) A Integers (pages 294 298) An integer is any number from this set of the whole numbers and their opposites: { 3, 2,, 0,, 2, 3, }. Integers that are greater than zero are positive integers. You can write

More information

Unit 7 The Number System: Multiplying and Dividing Integers

Unit 7 The Number System: Multiplying and Dividing Integers Unit 7 The Number System: Multiplying and Dividing Integers Introduction In this unit, students will multiply and divide integers, and multiply positive and negative fractions by integers. Students will

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

Name Partners Date. Energy Diagrams I

Name Partners Date. Energy Diagrams I Name Partners Date Visual Quantum Mechanics The Next Generation Energy Diagrams I Goal Changes in energy are a good way to describe an object s motion. Here you will construct energy diagrams for a toy

More information