Let s start with the definition of an equation.
|
|
- Eleanore Chrystal Goodman
- 7 years ago
- Views:
Transcription
1 2 MODULE 1. LINEAR EQUATIONS AND INEQUALITIES 1a Solving Equations Solving Equations: One Step Let s start with the definition of an equation. Equation. An equation is a mathematical statement that equates two mathematical expressions. The key difference between a mathematical expression and an equation is the presence of an an equals sign. So, for example, 2x +3, x (3 2x), and 2(y +3) 3(1 y) are mathematical expressions, while 2x +3=0, x (3 2x) =, and 2(y +3) 3(1 y) = 11 are equations. Note that each of the equations contain an equals sign, but the expressions do not. Next we have the definition of a solution of an equation. What it Means to be a Solution. A solution of an equation is a numerical value that satisfies the equation. That is, when the variable in the equation is replaced by the solution, a true statement results. Show that 2 is a solution of the equation 2y + 3 = 7. EXAMPLE 1. Show that 8 is a solution of the equation x 12 =. Solution. Substitute 8 for x in the given equation and simplify. x 12 = The given equation = Substitute 8 for x. = Since the left- and right-hand sides of the last line are equal, this shows that when 8 is substituted for x in the equation a true statement results. Therefore, 8 is a solution of the equation. Equivalent Equations
2 1A. SOLVING EQUATIONS 3 Equivalent Equations. Two equations are equivalent if they have the same solution set. EXAMPLE 2. Are the equations x 3=6andx = 9 equivalent? Are the equations x =5and Solution. The number 9 is the only solution of the equation x 3 = 6. x 7 = 10 equivalent? Similarly, 9 is the only solution of the equation x =9. Thereforex 3=6and x = 9 have the same solution sets and are equivalent. Answer: No. Operations that Produce Equivalent Equations We now turn to operations that will produce equivalent equations. Adding the Same Quantity to Both Sides of an Equation. Adding the same quantity to both sides of an equation does not change the solution set. That is, if a = b, then adding c to both sides of the equation produces the equivalent equation a + c = b + c. Subtracting the Same Quantity from Both Sides of an Equation. Subtracting the same quantity to both sides of an equation does not change the solution set. That is, if a = b, then subtracting c from both sides of the equation produces the equivalent equation a c = b c. Wrap and Unwrap Suppose that you are wrapping a gift for your cousin. You perform the following steps in order. 1. Put the gift paper on. 2. Put the tape on.
3 MODULE 1. LINEAR EQUATIONS AND INEQUALITIES 3. Put the decorative bow on. When we give the wrapped gift to our cousin, he politely unwraps the present, undoing each of our three steps in inverse order. 1. Take off the decorative bow. 2. Take off the tape. 3. Take off the gift paper. This seemingly frivolous wrapping and unwrapping of a gift contains some deeply powerful mathematical ideas. Consider the mathematical expression x +. To evaluate this expression at a particular value of x, we would start with the given value of x, then 1. Add. Suppose we started with the number 7. If we add, we arrive at the following result: 11. Now, how would we unwrap this result to return to our original number? We would start with our result, then 1. Subtract. That is, we would take our result from above, 11, then subtract, which returns us to our original number, namely 7. Addition and Subtraction as Inverse Operations. Two extremely important observations: The inverse of addition is subtraction. If we start with a number x and add a number a, then subtracting a from the result will return us to the original number x. In symbols, x + a a = x. The inverse of subtraction is addition. If we start with a number x and subtract a number a, then adding a to the result will return us to the original number x. In symbols, x a + a = x. Solve for x: EXAMPLE 3. Solve x 7=12forx. x 6=
4 1A. SOLVING EQUATIONS 5 Solution: To undo the effect of subtracting 7, we add 7 to both sides of the equation. x 7 = 12 x 7+7=12+7 x =19 Adding 7 to both sides of the equation produces an equivalent equation. On the left, adding 7 undoes the effect of subtracting 7 and returns x. On the right, = 19. Therefore, the solution of the equation is 19. Check: To check, substitute the solution 19 into the original equation. x 7 = =12 Substitute 19 for x. 12 = 12 The fact that the last line of the check is a true statement guarantees that 19 is a solution of x 7=12. Answer: 10 In the solution of Example 3, we use the concept of the inverse. If we start with x, subtract 7, then add 7, we are returned to the number x. In symbols, x 7+7=x. We are returned to x because subtracting 7 and adding 7 are inverse operations of one another. That is, whatever one does, the other undoes.
5 6 MODULE 1. LINEAR EQUATIONS AND INEQUALITIES Solve for x: x = 3 5 EXAMPLE. Solve x = 1 for x. 2 Solution: To undo the effect of adding 2/3, we subtract 2/3 from both sides of the equation. x = 1 2 x = Subtracting 2/3 from both sides produces an equivalent equation. x = x = 1 6 Therefore, the solution of the equation is 1/6. On the left, subtracting 2/3 undoes the effect of adding 2/3 and returns x. On the right, make equivalent fractions with a common denominator. 3 Subtract: 6 6 = 1 6 More Operations That Produce Equivalent Equations Here are two more operations that produce equivalent equations. Multiplying Both Sides of an Equation by a Nonzero Quantity. Multiplying both sides of an equation by a nonzero quantity does not change the solution set. That is, if a = b, and c 0, then multiplying both sides of the equation by c produces the equivalent equation ac = bc. Dividing Both Sides of an Equation by a Nonzero Quantity. Dividing both sides of an equation by a nonzero quantity does not change the solution set. That is, if a = b, and c 0, then dividing both sides of the equation by c produces the equivalent equation a c = b c.
6 1A. SOLVING EQUATIONS 7 Like addition and subtraction, multiplication and division are inverse operations. Multiplication and Division as Inverse Operations. Two extremely important observations: The inverse of multiplication is division. If we start with a number x and multiply by a number a, then dividing the result by the number a returns us to the original number x. In symbols, a x a = x. The inverse of division is multiplication. If we start with a number x and divide by a number a, then multiplying the result by the number a returns us to the original number x. In symbols, a x a = x. EXAMPLE 5. Solve 2.1x = 0.2 for x. Solve for x: Solution: To undo the effect of multiplying by 2.1, we divide both sides of the equation by x = x = x 2.1 = x = 2 Dividing both sides by 2.1 produces an equivalent equation. On the left, dividing by 2.1 undoes the effect of multiplying by 2.1 and returns x. On the right, divide: 0.2/( 2.1) = 2. Therefore, the solution of the equation is 2. Check: To check, substitute the solution 2 into the original equation. 2.1x = ( 2) = 0.2 Substitute 2 for x. 0.2 = 0.2 On the left, multiply: 2.1( 2) = 0.2 The fact that the last line of the check is a true statement guarantees that 2 is a solution of 2.1x = 0.2. Answer: 0.02
7 8 MODULE 1. LINEAR EQUATIONS AND INEQUALITIES Solving Equations: Clearing Fractions and Decimals In this section we introduce techniques that clear fractions and decimals from equations. When clearing fractions from an equation, you will need to simplify products like the ones posed in the following examples. Example?? shows all of the steps involved in arriving at the answer. Again, the goal in this section is to perform this calculation mentally, so we just Divide 7 into 28 to get, then multiply by 6 to get 2. ( ) x =2x Not only does this approach allow us to write down the answer without doing any work, the numerical calculations involve smaller numbers. You need to practice this mental calculation until you can write down the answer without writing down any steps. ( ) 5 EXAMPLE 6. Simplify: 27 9 x. Solution: Divide 9 into 27 to get 3, then multiply 3 by 5 to get 15. ( ) x =15x Simplify: Answer: 27x ( ) x Clearing Fractions Now that we ve done the required fraction work, we can now concentrate on clearing fractions from an equation. Clearing fractions from an equation. To clear fractions from an equation, multiply both sides of the equation by the least common denominator. EXAMPLE 7. Solve for x: x = 1 2. Solve for x: x 3 = 1 2
8 1A. SOLVING EQUATIONS 9 Solution: The common denominator for 2/3 and 1/2 is 6. We begin by multiplying both sides of the equation by 6. x = 1 ( 2 6 x + 2 ) ( ) 1 =6 3 2 ( ) ( ) 2 1 6x +6 =6 3 2 Multiply both sides by 6. On the left, distribute the 6. To simplify 6(2/3), you have two choices. You can multiply 6 and 2 to get 12, then divide 12 by 3 to get. Or you can divide 3 into 6 to get 2, then multiply 2 by 2 to get. Either way, 6(2/3) =. Similarly, 6(1/2) = 3. 6x +=3 Multiply: 6 ( ) ( ) 2 1 =,6 = Note that the fractions are now cleared from the equation. To isolate terms containing x on one side of the equation, subtract from both sides of the equation. 6x + =3 6x = 1 Subtract from both sides. To undo multiplying by 6, divide both sides by 6. 6x 6 = 1 6 x = 1 6 Divide both sides by 6. r x: 3 7 x = 3 2 EXAMPLE 8. Solve for x: 5 x = 3. Solution: The common denominator for /5 and /3 is 15. We begin by multiplying both sides of the equation by ( 5 x 5 x = ) 3 =15 ( 3 ) Multiply both sides by 15.
9 10 MODULE 1. LINEAR EQUATIONS AND INEQUALITIES To simplify 15(/5), you have two choices. Multiply 15 and to get 60, then divide 60 by 5 to get 12. Or you can divide 5 into 15 to get 3, then multiply 3 by to get 12. Either way, 15(/5) = 12. Similarly, 15( /3) = 20 12x = 20 Multiply. To undo multiplying by 12, we divide both sides by x 12 = x = 5 3 Divide both sides by 12. Reduce to lowest terms. Check: To check, substitute 5/3 for x in the original equation. 5 x = ( 3 5 ) = = 3 3 = 3 Substitute 5/3forx. Multiply numerators and denominators. Reduce. The fact that the last line is a true statement guarantees the 5/3 is a solution of the equation 5 x = 3. Answer: x = 7 2 EXAMPLE 9. Solve for x: 2x 3 3 = 1 2 3x. Solve for x: Solution: The common denominator for 2x/3, 3/, 1/2, and 3x/ is 12. We begin by multiplying both sides of the equation by ( 2x 3 2x 3 3 = 1 2 3x ( 2x ) ( 1 =12 2 3x ) ) ( ) ( ) = ( ) 3x Multiply both sides by 12. Distribute the 12 on each side. 5x = To simplify 12(2x/3), you have two choices. Multiply 12 by 2x to get 2x, then divide 2x by 3 to get 8x. Or you can divide 3 into 12 to get, then
10 1A. SOLVING EQUATIONS 11 multiply by 2x to get 8x. Either way, 12(2x/3) = 8x. Simlarly, 12(3/) = 9, 12(1/2) = 6, and 12(3x/) = 9x. 8x 9=6 9x Multiply. Note that the fractions are now cleared from the equation. We now need to isolate terms containing x on one side of the equation. To remove the term 9x from the right-hand side, add 9x to both sides of the equation. 8x 9+9x =6 9x +9x 17x 9=6 Add 9x to both sides. To remove the term 9 from the left-hand side, add 9 to both sides of the equation. 17x 9+9=6+9 17x = 15 Add 9 to both sides. Finally, to undo multiplying by 17, divide both sides of the equation by x 17 = 15 7 x = Divide both sides by 17. Clearing Decimals from an Equation Multiplying by the appropriate power of ten will clear the decimals from an equation. However, first note the following: 10(1.235) = place to the right. Multiplying by 10 moves the decimal point one 100(1.235) = Multiplying by 100 moves the decimal point two places to the right. 1000(1.235) = Multiplying by 1000 moves the decimal point three places to the right. Note the pattern: The number of zeros in the power of ten determines the number of places to move the decimal point. So, for example, if we multiply by 1,000,000, which has six zeros, this will move the decimal point six places to the right.
11 12 MODULE 1. LINEAR EQUATIONS AND INEQUALITIES Solve for x: EXAMPLE 10. Solve for x: 2.3x 1.25 = 0.0x x =2.2 Solution: The first term of 2.3x 1.25 = 0.0x has one decimal place, the second term has two decimal places, and the third and final term has two decimal places. At a minimum, we need to move each decimal point two places to the right in order to clear the decimals from the equation. Consequently, we multiply both sides of the equation by x 1.25 = 0.0x 100(2.3x 1.25) = 100(0.0x) Multiply both sides by (2.3x) 100(1.25) = 100(0.0x) Distribute the x 125 = x Multiplying by 100 moves all decimal points two places to the right. Note that the decimals are now cleared from the equation. We must now isolate all terms containing x on one side of the equation. To remove the term x from the right-hand side, subtract x from both sides of the equation. 230x 125 x =x x 226x 125 = 0 Subtract x from both sides. Simplify. To remove 125 from the left-hand side, add 125 to both sides of the equation. 226x = x = 125 Add 125 to both sides. Finally, to undo multiplying by 226, divide both sides by x 226 = x = Divide both sides by 226. Simplify.
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 informationPart 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 informationSolving Linear Equations in One Variable. Worked Examples
Solving Linear Equations in One Variable Worked Examples Solve the equation 30 x 1 22x Solve the equation 30 x 1 22x Our goal is to isolate the x on one side. We ll do that by adding (or subtracting) quantities
More informationMaths Workshop for Parents 2. Fractions and Algebra
Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)
More informationSolving Equations by the Multiplication Property
2.2 Solving Equations by the Multiplication Property 2.2 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the multiplication property to solve equations. Find the mean
More informationCOLLEGE ALGEBRA 10 TH EDITION LIAL HORNSBY SCHNEIDER 1.1-1
10 TH EDITION COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER 1.1-1 1.1 Linear Equations Basic Terminology of Equations Solving Linear Equations Identities 1.1-2 Equations An equation is a statement that two expressions
More informationChapter 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 information2.3 Solving Equations Containing Fractions and Decimals
2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions
More information2.3. 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 information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More informationSolving Systems of Two Equations Algebraically
8 MODULE 3. EQUATIONS 3b Solving Systems of Two Equations Algebraically Solving Systems by Substitution In this section we introduce an algebraic technique for solving systems of two equations in two unknowns
More informationLinear 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 informationLinear Equations in One Variable
Linear Equations in One Variable MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this section we will learn how to: Recognize and combine like terms. Solve
More informationDefinition 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 informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationAlgebraic 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 informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationFractions and Linear Equations
Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps
More informationMultiplying Fractions
. Multiplying Fractions. OBJECTIVES 1. Multiply two fractions. Multiply two mixed numbers. Simplify before multiplying fractions 4. Estimate products by rounding Multiplication is the easiest of the four
More information1.4 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 informationPreliminary Mathematics
Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and
More informationMBA Jump Start Program
MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right
More informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationSection 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 informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationMultiplying 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 informationSequences. A sequence is a list of numbers, or a pattern, which obeys a rule.
Sequences A sequence is a list of numbers, or a pattern, which obeys a rule. Each number in a sequence is called a term. ie the fourth term of the sequence 2, 4, 6, 8, 10, 12... is 8, because it is the
More informationSolution Guide Chapter 14 Mixing Fractions, Decimals, and Percents Together
Solution Guide Chapter 4 Mixing Fractions, Decimals, and Percents Together Doing the Math from p. 80 2. 0.72 9 =? 0.08 To change it to decimal, we can tip it over and divide: 9 0.72 To make 0.72 into a
More information1.6 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 informationIV. ALGEBRAIC CONCEPTS
IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other
More informationRULE 1: Additive Identity Property
RULE 1: Additive Identity Property Additive Identity Property a + 0 = a x + 0 = x If we add 0 to any number, we will end up with the same number. Zero is represented through the the green vortex. When
More informationBalancing 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 information3.1. 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 informationCAHSEE 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 informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More information0.8 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 informationEquations Involving Fractions
. Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation
More informationFactors Galore C: Prime Factorization
Concept Number sense Activity 4 Factors Galore C: Prime Factorization Students will use the TI-73 calculator s ability to simplify fractions to find the prime factorization of a number. Skills Simplifying
More informationCORE Assessment Module Module Overview
CORE Assessment Module Module Overview Content Area Mathematics Title Speedy Texting Grade Level Grade 7 Problem Type Performance Task Learning Goal Students will solve real-life and mathematical problems
More information3 cups ¾ ½ ¼ 2 cups ¾ ½ ¼. 1 cup ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼
cups cups cup Fractions are a form of division. When I ask what is / I am asking How big will each part be if I break into equal parts? The answer is. This a fraction. A fraction is part of a whole. The
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationLearning Objectives for Section 1.1 Linear Equations and Inequalities
Learning Objectives for Section 1.1 Linear Equations and Inequalities After this lecture and the assigned homework, you should be able to solve linear equations. solve linear inequalities. use interval
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationFree Pre-Algebra Lesson 55! page 1
Free Pre-Algebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
More informationFractions to decimals
Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}
More informationAlgebra 1. Practice Workbook with Examples. McDougal Littell. Concepts and Skills
McDougal Littell Algebra 1 Concepts and Skills Larson Boswell Kanold Stiff Practice Workbook with Examples The Practice Workbook provides additional practice with worked-out examples for every lesson.
More informationLESSON PLANS FOR PERCENTAGES, FRACTIONS, DECIMALS, AND ORDERING Lesson Purpose: The students will be able to:
LESSON PLANS FOR PERCENTAGES, FRACTIONS, DECIMALS, AND ORDERING Lesson Purpose: The students will be able to: 1. Change fractions to decimals. 2. Change decimals to fractions. 3. Change percents to decimals.
More informationLinear 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 informationAccuplacer Arithmetic Study Guide
Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how
More informationPERCENTS. Percent means per hundred. Writing a number as a percent is a way of comparing the number with 100. For example: 42% =
PERCENTS Percent means per hundred. Writing a number as a percent is a way of comparing the number with 100. For example: 42% = Percents are really fractions (or ratios) with a denominator of 100. Any
More informationPREPARATION FOR MATH TESTING at CityLab Academy
PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST
More informationIntegrating algebraic fractions
Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationRational 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 informationNumerator Denominator
Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3
More information3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes
Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same
More informationGCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1.
GCSE MATHEMATICS 43602H Unit 2: Number and Algebra (Higher) Report on the Examination Specification 4360 November 2014 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright
More informationChapter 1: Order of Operations, Fractions & Percents
HOSP 1107 (Business Math) Learning Centre Chapter 1: Order of Operations, Fractions & Percents ORDER OF OPERATIONS When finding the value of an expression, the operations must be carried out in a certain
More informationMATH-0910 Review Concepts (Haugen)
Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,
More informationQuestion 2: How do you solve a matrix equation using the matrix inverse?
Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients
More informationFRACTIONS COMMON MISTAKES
FRACTIONS COMMON MISTAKES 0/0/009 Fractions Changing Fractions to Decimals How to Change Fractions to Decimals To change fractions to decimals, you need to divide the numerator (top number) by the denominator
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationHFCC Math Lab Arithmetic - 4. Addition, Subtraction, Multiplication and Division of Mixed Numbers
HFCC Math Lab Arithmetic - Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers.
More informationPrentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)
New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct
More information7. 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 informationThis 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 informationExponents, Radicals, and Scientific Notation
General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationSolve 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 informationNo Solution Equations Let s look at the following equation: 2 +3=2 +7
5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are
More informationMethod 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 informationSimplifying Algebraic Fractions
5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions
More informationFactor Diamond Practice Problems
Factor Diamond Practice Problems 1. x 2 + 5x + 6 2. x 2 +7x + 12 3. x 2 + 9x + 8 4. x 2 + 9x +14 5. 2x 2 7x 4 6. 3x 2 x 4 7. 5x 2 + x -18 8. 2y 2 x 1 9. 6-13x + 6x 2 10. 15 + x -2x 2 Factor Diamond Practice
More informationAlgebra I Teacher Notes Expressions, Equations, and Formulas Review
Big Ideas Write and evaluate algebraic expressions Use expressions to write equations and inequalities Solve equations Represent functions as verbal rules, equations, tables and graphs Review these concepts
More informationAbsolute 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 informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationParamedic Program Pre-Admission Mathematics Test Study Guide
Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page
More information3.3 Addition and Subtraction of Rational Numbers
3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More information1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient
Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear
More information6-3 Solving Systems by Elimination
Warm Up Simplify each expression. 1. 2y 4x 2(4y 2x) 2. 5(x y) + 2x + 5y Write the least common multiple. 3. 3 and 6 4. 4 and 10 5. 6 and 8 Objectives Solve systems of linear equations in two variables
More information1.2 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 informationCalculator Worksheet--page 1
Calculator Worksheet--page 1 Name On this worksheet, I will be referencing keys that are on the TI30Xa. If you re using a different calculator, similar keys should be there; you just need to fi them! Positive/Negative
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More information5.1 Simple and Compound Interest
5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?
More informationFlorida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower
Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including
More informationPRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b.
PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.
More informationSolving Exponential Equations
Solving Exponential Equations Deciding How to Solve Exponential Equations When asked to solve an exponential equation such as x + 6 = or x = 18, the first thing we need to do is to decide which way is
More informationLet 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 informationIndices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková
Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead
More informationMATH 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 information6.4 Logarithmic Equations and Inequalities
6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.
More informationConversions between percents, decimals, and fractions
Click on the links below to jump directly to the relevant section Conversions between percents, decimals and fractions Operations with percents Percentage of a number Percent change Conversions between
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationSolving 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 informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More information