Math: Algebra Skills Flip Chart

Copyright © NewPath Learning. All rights reserved. www.newpathlearning.com Charts Charts Algebra Skills Algebra Skills Curriculum Mastery Flip Charts Combine Essential Math Skills with Hands-On Review! ® 33-6002 \|xiBAHBDy01266lz[ Sturdy, Free-Standing Design, Perfect for Learning Centers! Reverse Side Features Questions, Math Problems, Vocabulary Review & more!

Phone: 800-507-0966 • Fax: 800-507-0967 www.newpathlearning.com NewPath Learning® products are developed by teachers using research-based principles and are classroom tested. The company’s product line consists of an array of proprietary curriculum review games, workbooks, posters and other print materials. All products are supplemented with web-based activities, assessments and content to provide an engaging means of educating students on key, curriculum-based topics correlated to applicable state and national education standards. Copyright © 2009 NewPath Learning. All Rights Reserved. Printed in the United States of America. Curriculum Mastery® and NewPath Learning® are registered trademarks of NewPath Learning LLC. Math Curriculum Mastery® Flip Charts provide comprehensive coverage of key standards-based curriculum in an illustrated format that is visually appealing, engaging and easy to use. Curriculum Mastery® Flip Charts can be used with the entire classroom, with small groups or by students working independently. Each Math Curriculum Mastery® Flip Chart Set features • 10 double-sided laminated charts covering grade-level specific curriculum content on one side plus write-on/wipe-off charts on reverse side for student use or for small-group instruction. • Built-in sturdy free-standing easel for easy display • Spiral bound for ease of use • Activity Guide with black-line masters of the charts for students to fill-in, key vocabulary terms, corresponding quiz questions for each chart, along with answers Ideal for • Learning centers • In class instruction for interactive presentations and demonstrations • Hands-on student use • Stand alone reference for review of key science concepts • Teaching resource to supplement any program HOW TO USE Classroom Use Each Curriculum Mastery® Flip Chart can be used to graphically introduce or review a topic of interest. Side 1 of each Flip Chart provides graphical representation of key concepts in a concise, grade appropriate reading level for instructing students. The reverse Side 2 of each Flip Chart allows teachers or students to fill in the answers and summarize key concepts. Note: Be sure to use an appropriate dry-erase marker and to test it on a small section of the chart prior to using it. The Activity Guide included provides a black-line master of each Flip Chart which students can use to fill in before, during, or after instruction. On the reverse side of each black-line master are questions corresponding to each Flip Chart topic which can be used as further review or as a means of assessment. While the activities in the guide can be used in conjunction with the Flip Charts, they can also be used individually for review or as a form of assessment or in conjunction with any other related assignment. Learning Centers Each Flip Chart provides students with a quick illustrated view of grade-appropriate curriculum concepts. Students may use these Flip Charts in small group settings along with the corresponding activity pages contained in the guide to learn or review concepts already covered in class. Students may also use these charts as reference while playing the NewPath’s Curriculum Mastery® Games. Independent student use Students can use the hands-on Flip Charts to practice and learn independently by first studying Side 1 of the chart and then using Side 2 of the chart or the corresponding graphical activities contained in the guide to fill in the answers and assess their understanding. Reference/Teaching resource Curriculum Mastery® Charts are a great visual supplement to any curriculum or they can be used in conjunction with NewPath’s Curriculum Mastery® Games. Chart # 1: Chart # 2: Chart # 3: Chart # 4: Chart # 5: Chart # 6: Chart # 7: Chart # 8: Chart # 9: Chart #10: Introduction to Algebra Properties of Addition & Multiplication Introduction to Integers Adding & Subtracting Integers Multiplying & Dividing Integers Solving Equations Multistep Equations Inequalities Introduction to Functions Systems of Equations

Algebraic Expressions Translating word phrases into algebraic expressions Evaluating algebraic expressions • An algebraic expression is a mathematical phrase consisting of variables, numbers, and operations. Solving Equations • An equation is a mathematical sentence which shows that two expressions or values are equal. • Think of an equation as a balance scale. • To evaluate an algebraic expression, replace the variable with a number • Replace m with 4 in the expression The letter m is a variable which represents an unknown number. 3 x 2m is an algebraic expression evaluate m + 5 for m = 4 m + 5 4 + 5 = 9 Operation Word Phase Algebraic Expression Addition m + 9 Subtraction m – 9 Multiplication 2m + 9 2 x m 2 • m 2m Division • A number plus 9 • The sum of m and 9 • 9 more than a number • A number increased by 9 • The difference between a number and 9 • A number minus 9 • 9 less than a number • A number decreased by 9 • 9 more than twice a number • 2 x a number • A number multiplied by 2 • The product of 2 and a number • A number divided by 9 • 9 divided by a number • The quotient of a number and 9 m 9 9 m m 9 28 = m + 16 equal sign • What number plus 16 equals 28? Solve 16 + m = 28 Solution m = 12 12 + 16 = 28 28 = 28 • Solve the equation to find the value of the variable that makes the equation true. The value of the variable is called the solution. Examples of Algebraic Expressions Addition Subtraction Multiplication Division 6 + d, or d + 6 c - 6 6 x b, or 6b a ÷ 6, or a 6 expression expression 28 16 m © Copyright NewPath Learning. All Rights Reserved. 93-4507 www.newpathlearning.com Introduction to Algebra

Algebraic Expressions Translating word phrases into algebraic expressions Evaluating algebraic expressions • An algebraic expression is a mathematical phrase consisting of variables, numbers, and operations. Solving Equations • An equation is a mathematical sentence which shows that two expressions or values are equal. • Think of an equation as a balance scale. • To evaluate an algebraic expression, replace the variable with a number • Replace m with in the expression The letter m is a which represents an unknown number. 3 x 2m is an evaluate m + 5 for m = 4 m + 5 4 + 5 = 10 Operation Word Phase Algebraic Expression Addition Subtraction Multiplication Division • A number plus 9 • The sum of m and 9 • 9 more than a number • A number increased by 9 • The difference between a number and 9 • A number minus 9 • 9 less than a number • A number decreased by 9 • 9 more than twice a number • 2 x a number • A number multiplied by 2 • The product of 2 and a number • A number divided by 9 • 9 divided by a number • The quotient of a number and 9 • What number plus 16 equals 28? Solve 16 + m = 28 Solution m = 12 + 16 = 28 = 28 • Solve the equation to find the value of the variable that makes the equation true. The value of the variable is called the solution. Examples of Algebraic Expressions Addition Subtraction Multiplication Division 28 = m + 16 equal sign expression expression 28 16 m Key Vocabulary Terms • algebra • algebraic expression • difference • equation • increase • more • plus • product • quotient • solution • sum • variable © Copyright NewPath Learning. All Rights Reserved. 93-4507 www.newpathlearning.com Introduction to Algebra \|xiBAHBDy01648pzY

Addition Properties Property of Multiplication & Addition Multiplication Properties • Addition and multiplication properties help us solve basic algebraic equations. Property Description Formula Example Property Description Formula Example 6 + 4 = 4 + 6 8 + 0 = 8 You can add numbers in any order without changing their sum. The sum of any number and zero is equal to that number. You can group any of the numbers together without changing their sum. Associative (grouping) Identity Property of Zero Property Description Formula Example Property Description Formula Example You can add the numbers inside the parentheses first and then multiply, or multiply each number in the sum and then add. Property Description Formula Example Property Description Formula Example You can multiply numbers in any order without changing their sum. You can group any of the numbers together without changing their product. The product of any number and one is equal to that number. The product of any number and zero is zero. Associative (grouping) Identity Property of One Property of Zero Distributive a x b = b x a 9 x 6 = 6 x 9 a x 1 = a 4 x 1 = 4 a x 0 = 0 3 x 0 = 0 a + b = b + a a + 0 = a © Copyright NewPath Learning. All Rights Reserved. 93-4601 www.newpathlearning.com Properties of Addition & Multiplication

Addition Properties Property of Multiplication & Addition Multiplication Properties • Addition and multiplication properties help us solve basic . Property Description Formula Example Property Description Formula Example Associative (grouping) Identity Property of Zero Property Description Formula Example Property Description Formula Example Property Description Formula Example Property Description Formula Example Associative (grouping) Identity Property of One Property of Zero Distributive Key Vocabulary Terms • algebraic equation • distributive • associative property • identity property • commutative property of zero © Copyright NewPath Learning. All Rights Reserved. 93-4601 www.newpathlearning.com Properties of Addition & Multiplication \|xiBAHBDy01682tz]

x y -5 -4 -3 -2 -1 -10 -9 -8 -7 -6 1 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 0 (-2 , -6) (-1 , -2) (0 , 2) (1 , 6) (2 , 10) Graphing Linear Functions • A function is a rule that is performed on a number, called an input, to produce a result called an output. • x is the input • y is the output • By substituting a number for X, an output can be determined. • Graph the functions described by the equation: • Make a function table • Graph the ordered pairs on a coordinate plane. • Draw a line through the points which represent the values of the ordered pairs. • A graph of a linear function can be used to find the value of y for a given value of x. • A table can be used to show input and output values. y = 2x + 3 is an example of a function y = 4x + 2 y = 2x + 3 7 output 2 input Input 1 2 3 4 5 6 7 8 9 10 5 7 9 11 13 15 17 19 21 23 Output Output Input Function (4x + 2) Ordered Pairs (x, y) -2 -1 0 1 2 -6 -2 2 6 10 4 (-2) + 2 4 (-1) + 2 4 (0) + 2 4 (1) + 2 4 (2) + 2 (-2, -6) (-1, -2) (0, 2) (1, 6) (2, 10) © Copyright NewPath Learning. All Rights Reserved. 93-4603 www.newpathlearning.com Introduction to Functions

x y -5 -4 -3 -2 -1 -10 -9 -8 -7 -6 1 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 0 Graphing Linear Functions • A function is ________________________________ _____________________________________________ ____________________________________________ . • x is the input • y is the output • By substituting a number for , an can be determined. • Graph the functions described by the equation: • Make a function table • Graph the ordered pairs on a coordinate plane. • Draw a line through the points which represent the values of the ordered pairs. • A graph of a linear function can be used to find the value of for a given value of . • A table can be used to show input and output values. y = 2x + 3 is an example of a function y = 4x + 2 y = 2x + 3 output 2 input Input 1 2 3 4 5 6 7 8 9 10 Output Output Input Function (4x + 2) Ordered Pairs (x, y) -2 -1 0 1 2 Key Vocabulary Terms • coordinate plane • equation • function • input • linear function • ordered pair • output © Copyright NewPath Learning. All Rights Reserved. 93-4603 www.newpathlearning.com Introduction to Functions \|xiBAHBDy01671nzW

• Integers are a set of whole numbers and their opposites. • Positive integers are whole numbers greater than zero. • Negative integers are whole numbers less than zero. • The integer 0 is neither positive nor negative. • Integers have either a (+) positive or (–) negative sign, except zero, which has no sign. • The absolute value of an integer is the number of units from zero on a number line. • The absolute value of a number is always a positive number or zero. The symbol for absolute value is . • A number line can be used to order and compare integers. order from least to greatest • Read the numbers from left to right: • Opposite integers have the same absolute value. • A number line can be used to represent a set of integers. • The value of integers increases as you move to the right along a number line. And decreases in value as you move to the left. – 5 – 4 – 3 – 2 – 1 + 5 + 4 + 3 + 2 + 1 0 – 3 and 3 are opposites – 5 – 4 – 3 – 2 – 1 + 5 + 4 + 3 + 2 + 1 0 – 5 – 4 – 3 – 2 – 1 + 5 + 4 + 3 + 2 + 1 0 – 4 = 4 4 – 3, 0, 1, – 4, 2, 5 – 4, – 3, 0, 1, 2, 5 4 = 4 units 4 units Using Integers Integer Example – 350 + 100 – 35 – 10 350 feet below sea level 35 degrees below zero Loss of 10 yards in a football play Stock Market gain of 100 points Number Line negative integers positive integers Absolute Value Ordering Integers © C opyright NewPath Learning. All Rights Reserved. 93-4604 www.newpathlearning.com Introduction to Integers 40 50 60 70 80 90 100 110 ºF -50 -40 -30 -20 -10 0 10 20 30

• Integers are ________________ ____________________________ . • Positive integers are ________ ____________________________ . • Negative integers ___________ ____________________________ . • The integer is neither positive nor negative. • Integers have either a (+) positive or (–) negative sign, except zero, which has . • The absolute value of an integer is ______________________________________ _____________________________________ . • The absolute value of a number is always a or . The symbol for absolute value is . • A number line can be used to order and . order from least to greatest • Read the numbers from to : • integers have the same absolute value. • A can be used to represent a set of integers. • The value of integers as you move to the right along a number line. And in value as you move to the left. – 5 – 4 – 3 – 2 – 1 + 5 + 4 + 3 + 2 + 1 0 – 3 and 3 are opposites – 5 – 4 – 3 – 2 – 1 + 5 + 4 + 3 + 2 + 1 0 – 5 – 4 – 3 – 2 – 1 + 5 + 4 + 3 + 2 + 1 0 – 4 = 4 3 – 3, 0, 1, – 4, 5, 2 – 4, – 3, 0, 1, 2, 5 4 = Using Integers Integer Example – 350 + 100 – 35 – 10 350 feet below sea level 35 degrees below zero Loss of 10 yards in a football play Stock Market gain of 100 points Number Line Absolute Value Ordering Integers Key Vocabulary Terms • absolute value • integer • negative integer • number line • positive integer • whole number © Copyright NewPath Learning. All Rights Reserved. 93-4604 www.newpathlearning.com Introduction to Integers 40 50 60 70 80 90 100 110 ºF -50 -40 -30 -20 -10 0 10 20 30 \|xiBAHBDy01672kzU

– 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 • To add a positive integer, start at zero and move right on the number line, and move left to add a negative integer. move right 5 spaces from zero and then 2 more = 1 = 0 + = – 1 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 • To subtract a positive integer, move left on the number line, and right to subtract a negative number. 7 days 1 week (wk) 5 + 2 5 + 2 = 7 blue move right 4 spaces from zero and then left 6 move left 3 spaces from zero and then 7 more -3 + (-7) 4 + (-6) -3 + -7 = -10 4 + (-6) = -2 red red red red red red red red red red red red red red red red blue red red move right 5 spaces from zero and then left 3 5 – 3 5 – 3 = 2 blue blue blue blue blue blue blue move left 7 spaces from zero and then left 2 move left 6 spaces from zero and then right 4 -7 – 2 blue red red +2 5 4 -3 -6 5 – (-4) -3 -2 -7 -6 + (-7) blue blue blue -7 – 2 = - 9 7 -6 – (-4) = -2 -6 – (-4) red red red red red red red red red red red red red blue blue blue blue blue blue blue Adding Integers on a Number Line Subtracting Integers on a Number Line © Copyright NewPath Learning. All Rights Reserved. 93-4605 www.newpathlearning.com Adding & Subtracting Integers

• To add a positive integer, start at and move on the number line, and move to add a integer. move spaces from zero and then more = 1 = 0 + = – 1 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 – 5 – 4 – 3 – 2 – 1 – 10 – 9 – 8 – 7 – 6 5 4 3 2 1 10 9 8 6 0 7 • To subtract a positive integer, move on the number line, and to subtract a number. 5 + 2 5 + 2 = 7 blue -3 + (-7) 4 + (-6) -3 + -7 = -10 4 + (-6) = -2 red red red red red red red red red red red red red red red red blue red red 5 – 3 5 – 3 = 2 blue blue blue blue blue blue blue -7 – 2 blue red red blue blue blue -7 – 2 = - 9 7 -6 – (-4) = -2 -6 – (-4) red red red red red red red red red red red red red blue blue blue blue blue blue blue Adding Integers on a Number Line Subtracting Integers on a Number Line move spaces from zero and then more move spaces from zero and then move spaces from zero and then move spaces from zero and then move spaces from zero and then Key Vocabulary Terms • negative integer • number line • positive integer © Copyright NewPath Learning. All Rights Reserved. 93-4605 www.newpathlearning.com Adding & Subtracting Integers \|xiBAHBDy01663sz\

• Multiplication is repeated addition. • Multiplication with integers is commutative • Division is the inverse of multiplication • The same sign rules apply to multiplication and division of integers. – 5 – 4 – 3 – 2 – 1 – 10 – 12 – 11 – 9 – 8 – 7 – 6 5 4 3 2 1 10 11 12 9 8 6 0 7 4 x 3 means to add 3 four times: • Similarly, the product of 4 and – 3 means to add – 3 four times: Multiplying Integers Dividing Integers (-3) (-3) (-3) (-3) 4 groups of -3 = -12 3 + 3 + 3 + 3 –3 + (–3) + (–3) + (–3) –3 x (–2) = 6 5 x (–3) = –15 –15 ÷ (–3) = 5 –5 x (–3) = 15 15 ÷ (–3) = –5 6 ÷ (–3) = –2 –3 x 4 = –4 x (3) Example Rule You multiply integers just as you do whole numbers, except that you determine the sign of the product using these rules. If the factors have different signs, the product is negative. If the factors have the same signs, the product is positive. If one factor is zero, the product is zero. – 7 x 4 = – 28 – 4 x 0 = 0 – 8 x – 9 = 72 6 x 2 = 12 5 x – 3 = – 15 Example Rule • You cannot divide an integer by 0. If the dividend and divisor have different signs, the quotient is negative. If the dividend and divisor have the same signs, the quotient is positive. Zero divided by any integer equals 0. – 12 ÷ 4 = – 3 32 ÷ 4 = 8 15 ÷ (– 3) = – 5 – 21 ÷ (– 3) = 7 0 9 = 0 0 = 0 – 8 same signs product is positive different signs quotient is negative © C opyright NewPath Learning. All Rights Reserved. 93-4606 www.newpathlearning.com Multiplying & Dividing Integers

• Multiplication is • Multiplication with integers is commutative • Division is the inverse of multiplication • The same sign rules apply to multiplication and division of integers. – 5 – 4 – 3 – 2 – 1 – 10 – 12 – 11 – 9 – 8 – 7 – 6 5 4 3 2 1 10 11 12 9 8 6 0 7 4 x 3 means to times: • Similarly, the product of 4 and -3 means to times: Multiplying Integers Dividing Integers 4 groups of -3 = -12 3 + 3 + 3 + 3 –3 + (–3) + (–3) + (–3) –3 x (–2) = 6 5 x (–3) = –15 –15 ÷ (–3) = 5 –5 x (–3) = 15 15 ÷ (–3) = –5 6 ÷ (–3) = –2 –3 x 4 = –4 x ( –3 ) Example Rule You multiply integers just as you do whole numbers, except that you determine the sign of the product using these rules. If the factors have different signs, the product is negative. If the factors have the same signs, the product is positive. If one factor is zero, the product is zero. Example Rule • You cannot divide an integer by 0. If the dividend and divisor have different signs, the quotient is negative. If the dividend and divisor have the same signs, the quotient is positive. Zero divided by any integer equals 0. Key Vocabulary Terms • commutative • dividend • divisor • factor • integer • product • quotient • repeated addition © Copyright NewPath Learning. All Rights Reserved. 93-4606 www.newpathlearning.com Multiplying & Dividing Integers \|xiBAHBDy01674ozX

check check Division Equations Subtraction Equations Addition Equations check • An equation is a mathematical statement which shows that two quantities or values are equal. • Algebraic equations include at least one variable. • To solve an algebraic equation, you seek for the value of the variable that makes the equation a true statement. • The value of the variable is called the solution. • Since the equation shows subtraction, use the inverse operation, addition. • To get m by itself, add the same number (14) from both sides. • Solve the equation. • Substitute 43 for m in the equation. m = 43 • Think of an equation as a balance scale – both sides should be perfectly balanced. • When solving an equation, use inverse operations and keep the equation balanced by performing the exact thing to both sides of the equation. m + 16 = 28 • Since the equation shows addition, use the inverse operation, subtraction. • To get m by itself, subtract the same number (16) from both sides. • Solve the equation. • Substitute 12 for m in the equation. m + 16 = 28 28 = 28 m + 16 = 28 m + 16 = 28 12 + 16 = 28 -16 -16 m = 12 solve solve check solve solve • Since the equation shows multiplication, use the inverse operation, division. • Divide both sides by the same number (15). • Substitute 7 for m in the equation. 15m = 105 15m = 105 15(7) = 105 15m = 105 15 15 105 = 105 • Since the equation shows division, use the inverse operation, multiplication. • Multiply both sides by the same number (5). 5 m = 30 5 x x 5 5 m = 30 5 m = 30 5 150 = 30 m – 14 = 29 29 = 29 m – 14 = 29 m – 14 = 29 43 – 14 = 29 + 14 + 14 30 = 30 • Substitute 150 for m in the equation. ? ? ? ? m = 7 m = 150 Multiplication Equations © Copyright NewPath Learning. All Rights Reserved. 93-4608 www.newpathlearning.com Solving Equations 28 = m + 16 equal sign expression expression

\|xiBAHBDy01685kzU check check Division Equations Subtraction Equations Addition Equations check • An is a mathematical statement which shows that two quantities or values are . • Algebraic equations include at least one . • To solve an algebraic equation, you seek for the value of the that makes the equation a . • The value of the variable is called the . • Since the equation shows subtraction, use the inverse operation, addition. • To get m by itself, add the same number from both sides. • Solve the equation. • Substitute for m in the equation. m = 43 • Think of an equation as a balance scale – both sides should be perfectly . • When solving an equation, use and keep the by performing the exact thing to both sides of the equation. m + 16 = 28 • Since the equation shows addition, use the inverse operation, subtraction. • To get m by itself, subtract the same number from both sides. • Solve the equation. • Substitute for m in the equation. m + 16 = 28 28 = 28 m + 16 = 28 m + 16 = 28 12 + 16 = 28 -16 -16 m = 12 solve solve check solve solve • Since the equation shows multiplication, use the inverse operation, division. • Divide both sides by the same number . • Substitute for m in the equation. 15m = 105 15m = 105 15( ) = 105 15m = 105 15 15 105 = 105 • Since the equation shows division, use the inverse operation, multiplication. • Multiply both sides by the same number . 5 m = 30 5 x x 5 5 m = 30 5 m = 30 5 150 = 30 m – 14 = 29 29 = 29 m – 14 = 29 m – 14 = 29 43 – 14 = 29 + 14 + 14 30 = 30 • Substitute for m in the equation. ? ? ? ? m = 7 m = 150 Multiplication Equations © Copyright NewPath Learning. All Rights Reserved. 93-4608 www.newpathlearning.com Solving Equations 28 = m + 16 equal sign expression expression Key Vocabulary Terms • algebraic equation • solution • expression • value • inverse operation • variable

Solving Multistep Equations with Fractions Solving Multistep Equations with Like Terms 8y + 2 – 3y = 17 Solve: = 10 Solve: 5x – 5 7 = 10 = 70 5x – 5 7 5x – 5 • Follow the order of operations in reverse when solving equations with more than one operation (multistep equations). • Goals of solving multistep equations: • Place the variables on one side of the equal sign and the numbers on the other side. • Have the number in front of the variable equal to one. • Place the variables on the same side. • The number in front of the variable must be equal to one. • Two inverse operations are needed to solve the equation above – subtraction & division. Solving Two-Step Equations Using Division Solving Equations with Variables on Both Sides Solving Two-Step Equations Using Multiplication 3 x + 8 = 23 3 x + 8 = 15 x = 5 – 8 – 8 4 groups of -3 = -12 5n + 3 = 28 x – 5 = 20 5n = 25 = 5n 5 – 3 – 3 – 2 – 2 – 3x – 3x 4 • • 4 7 • • 7 + 5 + 5 + 5 + 5 0 9 x Subtract 3 from both sides of the equation. Multiply both sides by 4. Multiply both sides by 7. Divide both sides of the equation by 5. Divide both sides by 5. 25 5 1 4 = 25 x 1 4 = n 5n + 3 = 28 5 5x 5 75 5 = = = x 15 y = x 3 6 = x 100 Solve: x – 5 = 20 Solve: 1 4 Add 5 to both sides. 8y + 2 – 3y = 17 5y + 2 = 17 Multiply both sides by 7. Combine like terms (8y – 3y). Combine like terms (8y – 3y). Subtract 2 from both sides. Subtract 2 from both sides. 5y + 2 = 15 Divide both sides by 5. 5y 5 15 5 = 5x – 7 = 3x + 5 5x – 7 = 3x + 5 2x – 7 = 5 Solve: + 7 + 7 2x – 7 = 12 Add 7 to both sides. 2x 2 12 2 = Subtract 3x from both sides. Divide both sides by 2. © Copyright NewPath Learning. All Rights Reserved. 93-4701 www.newpathlearning.com Multistep Equations

\|xiBAHBDy01675lz[ Solving Multistep Equations with Fractions Solving Multistep Equations with Like Terms 8y + 2 – 3y = 17 Solve: = 10 Solve: 5x – 5 7 = 10 = 70 5x – 5 7 5x – 5 7 5x • Follow the order of operations in reverse when solving equations with more than one operation (multistep equations). • Goals of solving multistep equations: • _________________________________________ _________________________________________ • _________________________________________ _________________________________________ • Place the variables on the same side. • The number in front of the variable must be equal to . Solving Two-Step Equations Using Division Solving Equations with Variables on Both Sides Solving Two-Step Equations Using Multiplication 3 x + 8 = 23 3 x + 8 = 15 x = 5 – 8 – 8 5n + 3 = 28 x – 5 = 20 = = – 2 – 2 4 • • 4 7 • • 7 + 5 + 5 Subtract 3 from both sides of the equation. Multiply both sides by 4. Multiply both sides by 7. Divide both sides of the equation by 5. Divide both sides by 5. 1 4 = 25 x 1 4 = n = = x = y x = x 100 5n + 3 = 28 Solve: x – 5 = 20 Solve: 1 4 Add 5 to both sides. 8y + 2 – 3y = 17 5y + 2 = 17 Combine like terms (8y – 3y). Subtract from both sides. 5y + 2 = 15 Divide both sides by 5. 5y = 5x – 7 = 3x + 5 2x – 7 = 5 5x – 7 = 3x + 5 Solve: 2x = Add 7 to both sides. 2x = Subtract 3x from both sides. Divide both sides by 2. Key Vocabulary Terms • equation • inverse operation • multistep equation • operation • order of operations • variable • Two inverse operations are needed to solve the equation above – subtra ction & . . = © Copyright NewPath Learning. All Rights Reserved. 93-4701 www.newpathlearning.com Multistep Equations

• An inequality is a mathematical sentence that does not have an exact solution. Instead, a range of solutions will satisfy the inequality. • All the solutions of an inequality with more than one solution are called the solution set. • Inequalities are used in many real–world situations. An example is a driving speed sign with a number which tells you that your speed must be 65mph. • Graph each inequality separately. • Combine both graphs. • Solve an addition or subtraction inequality the same way as you would solve an equation. • When you multiply or divide both sides of an inequality by a negative integer, reverse the direction of the inequality symbol. Graphing Simple Inequalities Graphing Compound Inequalities Solving Inequalities by Adding or Subtracting Solving Inequalities by Multiplying or Dividing 4 groups of -3 = -12 0 9 x x + 3 5 x + 3 5 x 2 – 3 – 3 Example Meaning Symbol • greater than • more than • above • less than • fewer than • below • greater than or equal to • no less than • at least • less than or equal to • no more than • at most room capacity age – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 1 3 2 1 0 – 4 – 3 – 2 – 1 0 > 50 > > – > – 14 3 5 6 4 > – > > > –> – An inequality uses one of the following symbols, instead of an equal sign: An open circle is used when the variable is or a number. > > A closed circle is used when the variable is or a number. > > – – > x – 4 x 1 or > x – 2 y 1 > – > – x 1 > – x – 2 > x 2 > – > – > – > – – 2x 6 – 2x 6 – 2x 6 – 2 – 2 x – 3 > > > > © Copyright NewPath Learning. All Rights Reserved. 93-4702 www.newpathlearning.com Inequalities

• An inequality is a mathematical sentence that does not have an exact solution. Instead, a range of solutions will satisfy the inequality. • All the solutions of an inequality with more than one solution are called the so . • Inequal ities are used in many real–world situations. An example is a driving speed sign with a number which tells you that your speed must be 65mph. • Graph each inequality separately. • Combine both graphs. • Solve an addition or subtraction inequality the same way as you would solve an equation. • When you multiply or divide both sides of an inequality by a negative integer, revers ethe direction of the inequality . Graphing Simple Inequalities Graphing Compound Inequalities Solving Inequalities by Adding or Subtracting Solving Inequalities by Multiplying or Dividing Example Meaning Symbol • greater than • more than • above • less than • fewer than • below • greater than or equal to • no less than • at least • less than or equal to • no more than • at most room capacity age – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 1 3 2 1 0 – 4 – 3 – 2 – 1 0 > – An inequality uses one of the following symbols, instead of an equal sign: An op en circle is used when the variable is or a number. > > A c losed circle is used when the variable is or a number. > > – – > x – 4 x 1 or > x – 2 y 1 > – > – x 1 > – x – 2 > x + 3 5 x + 3 5 x 2 – 3 – 3 x 2 > – > – > – > – – 2x 6 – 2x 6 – 2x 6 x > > > © Copyright NewPath Learning. All Rights Reserved. 93-4702 www.newpathlearning.com Key Vocabulary Terms • closed circle • compound inequality • greater than • greater than or equal to • inequality • less than • less than or equal to • negative integer • open circle • simple inequality • solution • solution set • variable Inequalities \|xiBAHBDy01669kzU

• In algebra, two or more equations, called a system of equations, can be solved together. • A system of equations is a set of two or more equations with the same number of variables. • A solution of a system of equations is a set of values that are solutions which satisfy all the equations. y = -3 x – 2 2 x + y = 12 2 x + y = 12 12 – 2 x = -8 + 2 x 12 = -8 + 4 x y = 12 – 2 x y = 12 – 2 (5) y = 12 – 10 y = 2 4 x – 2 y = 16 – 2 x – 2 x y = 2 x – 2 y = -3 x – 2 y = -3 x – 2 y = 2 x – 2 -2 = -2 y = 2 x – 2 Solving Systems of Equations Identifying Solutions The Graphical Method x y -5 -4 -3 -2 -1 1 1 2 3 4 5 2 3 4 5 -1 -2 -3 -4 -5 0 • A system of equations can be solved by graphing all of the equations in the system. Is the ordered pair (0,-2) a solution of the system of equations below? • The solution of a system of equations is the point at which the lines intersect. • The solution set is (5,2) • The ordered pair (0,-2) is a solution of the system of equations. Solve the system of equations: Solve the system of equations: y = 2 x – 2 y = -3 x – 2 -2 = -3 (0) – 2 ? -2 = -2 -2 = 2 (0) – 2 + 2 x + 2 x + 8 + 8 4 x – 2 y = 16 -2 y = 16 – 4 x y = -8 + 2 x – 4 x – 4 x -2 -2 -2 • Solve both equations for y 20 = 4x 5 = x solution (0,-2) ? © Copyright NewPath Learning. All Rights Reserved. 93-4810 www.newpathlearning.com Systems of Equations

\|xiBAHBDy01687ozX • In algebra, two or more equations, called a , can be solved together. • A system of equations is ____________ _____________________________________ _____________________________________ . • A solution of a system of equations is _____________________________________ _____________________________________ . y = -3 x – 2 2 x + y = 12 2 x + y = 12 y = 4 x – 2 y = 16 y = 2 x – 2 y = -3 x – 2 y = -3 x – 2 y = 2 x – 2 y = 2 x – 2 Solving Systems of Equations Identifying Solutions The Graphical Method x y -5 -4 -3 -2 -1 1 1 2 3 4 5 2 3 4 5 -1 -2 -3 -4 -5 0 • A system of equations can be solved by graphing all of the equations in the system. Is the ordered pair (0,-2) a solution of the system of equations below? • The solution of a system of equations is the point at which the lines intersect. • The solution set is Solve the system of equations: Solve the system of equations: 4 x – 2 y = 16 • Solve both equations for y = x Key Vocabulary Terms • algebra • equation • ordered pair • solution set • system of equations • variable © Copyright NewPath Learning. All Rights Reserved. 93-4810 www.newpathlearning.com Systems of Equations

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