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Unit 1 SOLVING EQUATIONS AND EXPRESSIONS Unit 1 Vocabulary 1. absolute value 18. algebraic expression 2. addition property of equality 19. formula 3. additive inverse 20. grouping symbols 4. associative property of addition 5. associative property of 21. *identity property of addition multiplication 22. *identity property of multiplication 6. coefficient 23. inequality 7. commutative property of addition 24. inverse operation 8. commutative property of multiplication 25. like terms 9. constant (term) 26. multiplication property of equality 10. cross-product property 27. multiplicative inverse/reciprocal 11. distributive property 28. operation 12. *division by zero 13. division property of equality 29. order of operations 14. equation 30. *solve (an equation) 15. *equivalent equations 31. solution (of an equation in one 16. *equivalent expressions variable) 17. evaluate 32. *substitution Missing Terms division by zero: Division by zero is an undefined operation (you cannot find an answer) so it is disallowed. example: 12 ÷ 6 = 2 because 6 ∙2 =12 12 ÷ 0 = x would mean that 0 ∙ x = 12 But no value would work for x because 0 times any number is 0. So division by zero doesn't work. equivalent equations: equations that have the same solution. example: x - 6 = 5 and x - 11 = 0, both have the solution x = 11 equivalent expressions: expressions that simplify to an equal value when numbers are substituted for the variables of the expression. example: x + 4 + 2 =x + 6 (since 4 + 2 = 6) identity property of addition (a.k.a. additive identity): the sum of zero and x is x. example: ALGEBRA ARITHMETIC a+0=a 6+0=6 More Missing Terms identity property of multiplication: the product of 1 and x is x. example: ALGEBRA ARITHMETIC a∙1=a 6∙1=6 solve an equation: to find all value(s) of the variable(s) that satisfies to an equation, inequality, or a system of equations and/or inequalities. example: Solve for x. x+5=9 So, x = 4 substitution: a method used to evaluate algebraic expressions in which a variable, such as x or y, is replaced with its value. Example: Evaluate the expression for a=2 and b=5 a+b 2+5=7 Lesson 1 TRANSLATING BETWEEN WORDS AND ALGEBRA 1. GET A NOTES OUTLINE FROM THE C O R N E R O F T H E F R O N T TA B L E 2. COPY YOUR HW 3. S TA RT O N WA R M U P ( TO P O F O U T L I N E ) Operation terms: Draw a table with the following 7 column headings. Make your table have 5 rows below the heading. + - x ÷ = ( ) < > increased decreased of quotient is quantity is greater by by than more minus product to are sum is less than than add subtract times divided was difference is no more among than total less than were is at least/ most sum difference Writing Math These expressions all mean “2 times y”: 2y 2(y) 2•y (2)(y) 2×y (2)y Example 1: Translating from Algebra to Words Give two ways to write each algebra expression in words. A. 9 + r B. q – 3 the sum of 9 and r the difference of q and 3 9 increased by r 3 less than q C. 7m D. j ÷ 6 the product of m and 7 the quotient of j and 6 m times 7 j divided by 6 You Try! Example 1 Give two ways to write each algebra expression in words. 1a. 4 - n 1b. 4 decreased by n the quotient of t and 5 n less than 4 t divided by 5 1c. 9 + q 1d. 3(h) the sum of 9 and q the product of 3 and h q added to 9 3 times h Example 2A: Translating from Words to Algebra John types 62 words per minute. Write an expression for the number of words he types in m minutes. m represents the number of minutes that John types. 62 · m or 62m Think: m groups of 62 words Example 2B: Translating from Words to Algebra Roberto is 4 years older than Emily, who is y years old. Write an expression for Roberto’s age y represents Emily’s age. y+4 Think: “older than” means “greater than.” Example 2C: Translating from Words to Algebra Joey earns $5 for each car he washes. Write an expression for the number of cars Joey must wash to earn d dollars. d represents the total amount that Joey will earn. Think: How many groups of $5 are in d? You try! Example 2a Lou drives at 65 mi/h. Write an expression for the number of miles that Lou drives in t hours. t represents the number of hours that Lou drives. 65t Think: number of hours times rate per hour. You Try! Example 2b Miriam is 5 cm taller than her sister, who is m centimeters tall. Write an expression for Miriam’s height in centimeters. m represents Miriam’s sister's height in centimeters. m+5 Think: Miriam's height is 5 added to her sister's height. You Try! Example 2c Elaine earns $32 per day. Write an expression for the amount she earns in d days. d represents the amount of money Elaine will earn each day. 32d Think: The number of days times the amount Elaine would earn each day. What is an Algebraic Expression? Important: Expressions DO NOT have equal (=) signs Evaluating Expressions To evaluate an expression means to find its value. To evaluate an algebraic expression… 1. Substitute numbers for the variables in the expression 2. Apply order of operations. 3. Simplify the expression. Example 3: Evaluating an Algebraic Expression Evaluate each expression for a = 4, b =7, and c = 2. A. b – c b–c=7–2 Substitute 7 for b and 2 for c. =5 Simplify. B. ac ac = 4 ·2 Substitute 4 for a and 2 for c. =8 Simplify. You Try! Example 3 Evaluate each expression for m = 3, n = 2, and p = 9. a. mn mn = 3 · 2 Substitute 3 for m and 2 for n. =6 Simplify. b. p – n p–n=9–2 Substitute 9 for p and 2 for n. =7 Simplify. c. p ÷ m p÷m=9÷3 Substitute 9 for p and 3 for m. =3 Simplify. Problem Solving: Recycling Application Approximately eighty-five 20-ounce plastic bottles must be recycled to produce the fiberfill for a sleeping bag. Write an expression for the number of bottles needed to make s sleeping bags. The expression 85s models the number of bottles to make s sleeping bags. Recycling Application Continued Approximately eighty-five 20-ounce plastic bottles must be recycled to produce the fiberfill for a sleeping bag. Find the number of bottles needed to make 20, 50, and 325 sleeping bags. Evaluate 85s for s = 20, 50, and 325. s 85s To make 20 sleeping bags 1700 bottles are needed. 20 85(20) = 1700 To make 50 sleeping bags 4250 50 85(50) = 4250 bottles are needed. To make 325 sleeping bags 325 85(325) = 27,625 27,625 bottles are needed. Lesson 2 REVIEW OF INTEGER OPERATIONS AND EXPONENTS The absolute value of a number is the distance from zero on a number line. The absolute value of 5 is written as |5|. 5 units 5 units - 6 - 5 - 4 - 3 - 2 -1 0 1 2 3 4 5 6 |–5| = 5 |5| = 5 Review: Operations with Integers Adding Integers Same Sign (pos + pos or neg + neg)= Add, Keep the Sign Different Signs (pos + neg) = Subtract, Keep the “bigger” number’s sign (absolute value) Subtracting Integers Change subtraction to addition, change the sign of the 2nd number and follow addition rules Multiplying and Dividing Integers (same rules) Same Sign = Positive Different Signs = Negative Example 1A: Adding Real Numbers Add. y + (–2) for y = –6 y + (–2) = (–6) + (–2) First substitute –6 for y. When the signs are the same, (–6) + (–2) find the sum of the absolute values: 6 + 2 = 8. –8 Both numbers are negative, so the sum is negative. Example 1B: Adding Real Numbers When the signs of numbers are different, Add. find the difference of the absolute values: Use the sign of the number with the greater absolute value. The sum is negative. You Try! Example 1A Add. –5 + (–7) –5 + (–7) = 5 + 7 When the signs are the same, find the sum of the absolute values. 5 + 7 = 12 Both numbers are negative, so the sum –12 is negative. You Try! Example 1b Add. –13.5 + (–22.3) –13.5 + (–22.3) When the signs are the same, find the sum of the absolute values. 13.5 + 22.3 –35.8 Both numbers are negative so, the sum is negative. Check It Out! Example 1c Add. x + (–68) for x = 52 First substitute 52 for x. x + (–68) = 52 + (–68) When the signs of the numbers are different, find the difference of the absolute values. 68 – 52 Use the sign of the number with the greater absolute value. –16 The sum is negative. Example 2A: Subtracting Real Numbers Subtract. –6.7 – 4.1 –6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1. When the signs of the numbers are the same, find the sum of the absolute values: 6.7 + 4.1 = 10.8. = –10.8 Both numbers are negative, so the sum is negative. Example 2B: Subtracting Real Numbers Subtract. 5 – (–4) 5 − (–4) = 5 + 4 To subtract –4 add 4. 9 Find the sum of the absolute values. You Try! Example 2a Subtract. 13 – 21 13 – 21 To subtract 21 add –21. 13 + (–21) When the signs of the numbers are different, find the difference of the absolute values: 21 – 13 = 8. –8 Use the sign of the number with the greater absolute value. You Try! Example 2b Subtract. x – (–12) for x = –14 x – (–12) = –14 – (–12) First substitute –14 for x. –14 + (12) To subtract –12, add 12. When the signs of the numbers are different, find the difference of the absolute values: 14 – 12 = 2. Use the sign of the number with the greater –2 absolute value. Example 3: Multiplying and Dividing Signed Numbers Find the value of each expression. The product of two numbers A. with different signs is negative. –5 B. The quotient of two numbers 12 with the same sign is positive. Example 3C: Multiplying and Dividing Signed Numbers Find the value of the expression. First substitute for x. The quotient of two numbers with different signs is negative. You Try! Example 3 Find the value of each expression. A. 35 (–5) The quotient of two numbers –7 with different signs is negative. B. –11(–4) The product of two numbers 44 with the same sign is positive. C. –6x for x = 7 –6x = –6(7) First substitute 7 for x. = –42 The product of two numbers with different signs is negative. Properties of Zero Multiplication by Zero The product of any number and 0 is 0. WORDS 1 NUMBERS ·0=0 0(–17) = 0 3 ALGEBRA a·0=0 0·a=0 Properties of Zero Zero Divided by a Number The quotient of 0 and any nonzero number is 0. WORDS 0 2 NUMBERS =0 0÷ =0 6 3 0 ALGEBRA =0 0÷a=0 a Properties of Zero Division by Zero WORDS Division by 0 is undefined. 12 ÷ 0 –5 NUMBERS Undefined 0 a ALGEBRA a÷0 0 Undefined Lesson 3 ORDER OF OPERATIONS AND EVALUATING EXPRESSIONS Warm Up Simplify this expression. 5 + (8 – 6)2 ⋅ 3 Which operation do you perform first? Review: Evaluating Exponents An exponent is a number that tells how many times the base number is used as a factor. For example, 34 indicates that the base number 3 is used as a factor 4 times. To determine the value of 34, multiply 3*3*3*3 which would give the result 81. ** Be careful – many students get confused and instead of multiplying 3 times itself 4 times, they multiply 3 times 4 and get 12. This is incorrect! Exponents are written as a superscript number (ex. 34). When a number is written with an exponent, this is called exponential notation. Some important facts about exponents: Zero raised to any power is zero (ex. 05 = 0) One raised to any power is one (ex. 15 = 1) Any number raised to the zero power is one (ex. 70 = 1) Any number raised to the first power is that number (ex. 71 = 7) When a numerical or algebraic expression contains more than one operation symbol, the order of operations tells which operation to perform first. Using order of operations correctly, everyone should get the same answer. Order of Operations First: Perform operations inside grouping symbols. Second: Evaluate powers. Third: Perform multiplication and division from left to right. Fourth: Perform addition and subtraction from left to right. Grouping symbols include parentheses ( ), brackets [ ], and braces { }. If an expression contains more than one set of grouping symbols, evaluate the expression from the innermost set first. Helpful Hint The first letter of these words can help you remember the order of operations. Geez Grouping Excuse Symbols My Exponents Dear Multiply Aunt Divide Sally Add Subtract Example 1 Simplify each expression. A. 15 – 2 · 3 + 1 15 – 2 · 3 + 1 There are no grouping symbols. 15 – 6 + 1 Multiply. 10 Subtract and add from left to right. B. 12 – 32 + 10 ÷ 2 There are no grouping symbols. 12 – 32 + 10 ÷ 2 Evaluate powers. The 12 – 9 + 10 ÷ 2 exponent applies only to the 3. Divide. 12 – 9 + 5 Subtract & add from left to right. 8 Example 2: Evaluating Algebraic Expressions Evaluate the expression for the given value of x. 10 – x · 6 for x = 3 10 – x · 6 First substitute 3 for x. 10 – 3 · 6 Multiply. 10 – 18 Subtract. –8 Example 2B: Evaluating Algebraic Expressions Evaluate the expression for the given value of x. 42(x + 3) for x = –2 42(x + 3) First substitute –2 for x. 42(–2 + 3) Perform the operation 42(1) inside the parentheses. 16(1) Evaluate powers. 16 Multiply. You Try! Evaluate the expression for the given value of x. 14 + x2 ÷ 4 for x = 2 14 + x2 ÷ 4 14 + 22 ÷ 4 First substitute 2 for x. 14 + 4 ÷ 4 Square 2. 14 + 1 Divide. 15 Add. You Try! Evaluate the expression for the given value of x. (x · 22) ÷ (2 + 6) for x = 6 (x · 22) ÷ (2 + 6) (6 · 22) ÷ (2 + 6) First substitute 6 for x. (6 · 4) ÷ (2 + 6) Square two. (24) ÷ (8) Perform the operations inside the parentheses. 3 Divide. Fraction bars, radical symbols, and absolute-value symbols can also be used as grouping symbols. Remember that a fraction bar indicates division. Example 3: Simplifying Expressions with Other Grouping Symbols Simplify. 2(–4) + 22 The fraction bar acts as a grouping 42 – 9 symbol. Simplify the numerator and the denominator (separately) before dividing. Multiply to simplify the numerator. Evaluate the power in the denominator. Add to simplify the numerator. Subtract to simplify the denominator. Divide. Lesson 5 EQUIVALENT EXPRESSIONS What is equivalence? Equivalent means the same thing as equal, so when items are equivalent, their value is the same. When expressions are equivalent they simplify to an equal value when numbers are substituted for the variables. example: x + 4 + 2 =x + 6 (since 4 + 2 = 6) When equations are equivalent, they have the same solution. example: x - 6 = 5 and x - 11 = 0, both have the solution x = 11 QUIZ WEDNESDAY! Review the following for the quiz: Lesson 1 – Translating between words and Algebra Lesson 2 – Operations with Integers (Review from last year) Lesson 3 – Order of Operations and Evaluating Expressions (Review from last year) Lesson 4 – Equivalent Expressions Lesson Quiz: Part I Give two ways to write each algebraic expression in words. 1. j – 3 2. 4p 3. Mark is 5 years older than Juan, who is y years old. Write an expression for Mark’s age. Lesson Quiz: Part II Evaluate each expression for c = 6, d = 5, and e = 10. 4. e 5. c + d d Shemika practices basketball for 2 hours each day. 6. Write an expression for the number of hours she practices in d days. 7. Find the number of hours she practices in 5, 12, and 20 days. Lesson Quiz: Part III Use the following terms to complete the blanks below: variable, expression, constant, coefficient 8. 4x + 7 is an algebraic _________________. 9. The “4” in problem number 8 is the ________________. 10. The “x” in problem number 8 is the _________________. 11. The “7” in problem number 8 is the _________________. Lesson Quiz: Part I Give two ways to write each algebraic expression in words. 1. j – 3 The difference of j and 3; 3 less than j. 2. 4p 4 times p; The product of 4 and p. 3. Mark is 5 years older than Juan, who is y years old. Write an expression for Mark’s age. y + 5 Lesson Quiz: Part II Evaluate each expression for c = 6, d = 5, and e = 10. 4. e 2 5. c + d 11 d Shemika practices basketball for 2 hours each day. 6. Write an expression for the number of hours she practices in d days. 2d 7. Find the number of hours she practices in 5, 12, and 20 days. 10 hours; 24 hours; 40 hours Lesson Quiz: Part III Use the following terms to complete the blanks below: variable, expression, constant, coefficient expression 8. 4x + 7 is an algebraic _________________. coefficient 9. The “4” in problem number 8 is the ________________. variable 10. The “x” in problem number 8 is the _________________. constant 11. The “7” in problem number 8 is the _________________. Lesson 4 SIMPLIFYING EXPRESSIONS What is a SIMPLIFIED EXPRESSION? Simplified means to put something in a more compact, sometimes easier to read form ( basically…it makes it “simpler”. A simplified algebraic expression will… not have operations that have not been performed not have grouping symbols not have any like terms that are not combined The Commutative and Associative Properties of Addition and Multiplication allow you to rearrange an expression to simplify it. Example 1A: Using the Commutative and Associative Properties Simplify. Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. 11(5) 55 Example 1B: Using the Commutative and Associative Properties Simplify. 45 + 16 + 55 + 4 45 + 55 + 16 + 4 Use the Commutative Property. (45 + 55) + (16 + 4) Use the Associative Property to make groups of compatible numbers. (100) + (20) 120 Helpful Hint Compatible numbers help you do math mentally. Try to make multiples of 5 or 10. They are simpler to use when multiplying. You Try! Example 1a Simplify. Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. 21 You Try! Example 1b Simplify. 410 + 58 + 90 + 2 410 + 90 + 58 + 2 Use the Commutative Property. Use the Associative Property to make groups (410 + 90) + (58 + 2) of compatible numbers. (500) + (60) 560 You Try! Example 1c Simplify. 1 • 7•8 2 1 • 8•7 Use the Commutative Property. 2 1 (2 • 8 )7 Use the Associative Property to make groups of compatible numbers. 4 • 7 28 The Distributive Property is used with Addition to Simplify Expressions. The Distributive Property also works with subtraction because subtraction is the same as adding the opposite. Example 2A: Using the Distributive Property with Mental Math Write the product using the Distributive Property. Then simplify. 5(59) 5(50 + 9) Rewrite 59 as 50 + 9. 5(50) + 5(9) Use the Distributive Property. 250 + 45 Multiply. 295 Add. Example 2B: Using the Distributive Property with Mental Math Write the product using the Distributive Property. Then simplify. 8(33) 8(30 + 3) Rewrite 33 as 30 + 3. 8(30) + 8(3) Use the Distributive Property. 240 + 24 Multiply. 264 Add. You Try! Example 2a Write the product using the Distributive Property. Then simplify. 9(52) 9(50 + 2) Rewrite 52 as 50 + 2. 9(50) + 9(2) Use the Distributive Property. 450 + 18 Multiply. 468 Add. You Try! Example 2b Write the product using the Distributive Property. Then simplify. 12(98) 12(100 – 2) Rewrite 98 as 100 – 2. 12(100) – 12(2) Use the Distributive Property. 1200 – 24 Multiply. 1176 Subtract. Distributive Property…algebraically speaking To us the distributive property to simplify an algebraic expression… Multiply the number outside the parenthesis by each number inside the parenthesis SEPARATELY. Examples: 5(4 + x) (x - 2)4 ½(4x + 10) Punchline Worksheet 2.10 Get rid of parenthesis by applying distributive property…do not simplify further. Show all work on the back of the sheet. The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms. Like terms Constant 4x – 3x + 2 A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1. Coefficients 1x2 + 3x Like Terms Activity You need the sticky note at your table. Notice the options hanging around the room. Stick your term with a like term. Be able to defend why your term is like the one you chose. Example 3A: Combining Like Terms Simplify the expression by combining like terms. 72p – 25p 72p – 25p 72p and 25p are like terms. 47p Subtract the coefficients. Example 3B: Combining Like Terms Simplify the expression by combining like terms. A variable without a coefficient has a coefficient of 1. and are like terms. Write 1 as Add the coefficients. Example 3C: Combining Like Terms Simplify the expression by combining like terms. 0.5m + 2.5n 0.5m + 2.5n 0.5m and 2.5n are not like terms. 0.5m + 2.5n Do not combine the terms. You Try! Example 3 Simplify by combining like terms. 3a. 16p + 84p 16p + 84p 16p + 84p are like terms. 100p Add the coefficients. 3b. –20t – 8.5t2 –20t – 8.5t2 20t and 8.5t2 are not like terms. –20t – 8.5t2 Do not combine the terms. 3c. 3m2 + m3 3m2 + m3 3m2 and m3 are not like terms. 3m2 + m3 Do not combine the terms. Example 4: Simplifying Algebraic Expressions Simplify 14x + 4(2 + x). Justify each step. Procedure Justification 1. 14x + 4(2 + x) 2. 14x + 4(2) + 4(x) Distributive Property 3. 14x + 8 + 4x Multiply. Commutative Property 4. 14x + 4x + 8 5. (14x + 4x) + 8 Associative Property 6. 18x + 8 Combine like terms. You Try! Example 4a Simplify 6(x – 4) + 9. Justify each step. Procedure Justification 1. 6(x – 4) + 9 2. 6(x) – 6(4) + 9 Distributive Property 3. 6x – 24 + 9 Multiply. Combine like terms. 4. 6x – 15 Finish Punchline Worksheet 2.10 Since you already the used the distributive property to get rid of ( ) yesterday, today combine like terms to simplify the expression completely. Solve the riddle if you’d like to.