Clifton High School Mathematics Summer Workbook Algebra 1

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1 1 Clifton High School Mathematics Summer Workbook Algebra 1 Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature: Parent Signature: Please read the note to parents following this page.

3 CLIFTON HIGH SCHOOL MATHEMATICS DEPARTMENT 3 ALGEBRA I SUMMER WORKBOOK TOPICS COVERED Basic Vocabulary 1-1 Variables and Expressions (includes Order of Operations) 1-2 Solving Equations by Adding and Subtracting 1-3 Solving Equations by Multiplying and Dividing 1-4 Solving Multi-Step Equations 1-5 Solving Equations with Variables on Both Sides 1-6 Solving for a Variable 1-7 Solving Absolute Value Equations The Algebra 1 textbook can be accessed for reference over the summer. There are instructional videos to explain each lesson. Username: cwwms1 Password: clifton Study Island can also be accessed during the summer. Students should be familiar with the 8 th grade common core topics. All pages MUST show the work in order for the answers to be accepted. All work should be written neatly on a separate page. This booklet must be kept neat and in order and is to remain in your notebook as a reference guide. 3-Ring binders are required for 9 th grade Algebra. Completion of this booklet is required on the first day of the school year.

4 VOCABULARY: Match the given words to the correct definition. Write the answers on this page. absolute value equation GCF prime number sum base exponent integers product variable composite numbers expression LCM quotient < difference factors ordered pair rational number > 1) a mathematical sentence that contains an equal sign 4 2) made up of quantities and the operations performed on them 3) a symbol that is used to represent a number does not contain =,, <,, >, 4) used to locate points (x, y) in the coordinate plane 5) the solution to an addition problem 6) the solution to a subtraction problem 7) the solution to a multiplication problem 8) the solution to a division problem 9) whole numbers and their opposites { -3,-2,-1,0,1,2,3, } (symbol is Z) 10) a number that can be expressed in the form a/b, in which a and b are integers and b 0 (symbol is Q) 11) a number s distance from zero on the number line 12) the quantities that are multiplied in a multiplication expression 13) a whole number greater than one, with exactly 2 factors, 1 and itself 14) a whole number greater than 1 that has more than 2 factors 15) the greatest number that is a factor of two or more integers 16) the least positive integer that is divisible by each of 2 or more integers 17) the x in an expression of the form x n 18) the n in an expression of the form x n 19) the symbol for less than (2 3) 20) the symbol for greater than (3 2)

5 x-x 1-x 1-1 Variables and Expressions To translate words into algebraic expressions, find words like these that tell you the operation. + add subtract multiply divide sum difference product quotient more less times split increased decreased per ratio Kenny owns v video games. Stan owns 7 more video games than Kenny. Write an expression for the number of video games Stan owns. v represents the number of video games Kenny owns. v + 7 Think: The word more indicates addition. Order does not matter for addition. The expression 7 + v is also correct. Jenny is 12 years younger than Candy. Write an expression for Jenny s age if Candy is c years old. c represents Candy s age. The word younger means less, which indicates subtraction. c 12 Think: Candy is older, so subtract 12 from her age. Order does matter for subtraction. The expression 12 c is incorrect. 1. Jared can type 35 words per minute. Write an expression for the number of words he can type in m minutes. 2. Mr. O Brien s commute to work is 0.5 hour less than Miss Santos s commute. Write an expression for the length of Mr. O Brien s commute if Miss Santos s commute is h hours. 3. Mrs. Knighten bought a box of c cookies and split them evenly between the 25 students in her classroom. Write an expression for the number of cookies each student received. 4. Enrique collected 152 recyclable bottles, and Latasha collected b recyclable bottles. Write an expression for the number of bottles they collected altogether. 5. Tammy s current rent is r dollars. Next month it will be reduced by \$50. Write an expression for next month s rent in dollars. 1-6

6 x-x 1-x 1-1 The value of Variables and Expressions continued 9 depends on what number is placed in the box. Evaluate 9 when 20 is placed in the box In algebra, variables are used instead of boxes. Evaluate x 7 for x = 28. x Sometimes, the expression has more than one variable. Evaluate x + y for x = 6 and y = 2. x + y Evaluate 5 + when each number is placed in the box Evaluate each expression for x = 4, y = 6, and z = x y z Evaluate each expression for x = 2, y = 18, and z = x z 13. y x 14. y z 15. y x 16. xy 17. z x 1-7

7 x-x 1-x 1-2 Solving Equations by Adding or Subtracting Use counters to model solving equations. Solve x + 2 = 5. Using counters Using numbers x + 2 = 5 x + 2 = x + 0 = 3 x = 3 Check: Check: x + 2 = =? 5 5 =? 5 Solve the following by drawing counters. Check your answers. 1. x + 1 = = x + 2 Solve each equation. Check your answers. 3. x + 4 = = x x + 3 =

9 x-x 1-x 1-3 Solving Equations by Multiplying or Dividing Solve equations involving multiplication and division by performing the inverse operation. Solve x 5 = 4. Check: x 5 = 4 x 5 = 4 x is divided by =? 4 5 x 5 = 4 5 Multiply both sides by 5. 4 =? 4 5x 5 = 20 Simplify. x = 20 Solve 3x = 27. Check: 3x = 27 3x = 27 x is multiplied by 3. 3( 9) =? 27 3x 3 = 27 3 x = 9 Divide both sides by 3. Simplify. 27 =? 27 Circle the correct word in each sentence. Then solve the equation. 1. x 2 = m = 40 x is multiplied/divided by 2. m is multiplied/divided by 5. To solve, multiply/divide both sides by 2. To solve, multiply/divide both sides by 5. x = m = Solve each equation. Check your answers. 3. 2x = w = z =

10 x-x 1-x 1-3 Solving Equations by Multiplying or Dividing continued Equations with fractions can be solved by multiplying by the reciprocal. Solve 2 3 x = 8. Check: 2 3 x = x = 8 x is multiplied by (12) =? x = x = 24 2 x = 12 Multiply both sides by =? 8 Simplify. 8 =? 8 Solve 3 4 x = 2 5. Check: 3 4 x = x = x = x = 8 15 x = 8 15 x is multiplied by =? x is multiplied by =? 5 Simplify =? 5 Find the reciprocal Solve each equation. Check your answers x = = 3 5 x x =

11 x-x 1-x 1-4 Solving Two-Step and Multi-Step Equations When solving multi-step equations, first combine like terms on each side if possible. Then use inverse operations. 4x 3 = 15 Operations x is multiplied by 4. Then 3 is subtracted. Solve using Inverse Operations Add 3 to both sides. Then divide both sides by 4. x = 9 x is divided by 3. Then 2 is added. Add 2 to both sides. Then multiply both sides by 3. The order of the inverse operations is the order of operations in reverse. Solve 2x 7 + 3x = 13. Check: 2x + 3x 7 = 13 Group like terms together. 2x 7 + 3x = 13 5x 7 = 13 Add like terms. 2(4) 7 + 3(4) =? 13 5x 7 = 13 x is multiplied by 5. Then 7 is subtracted Add 7 to both sides =? 13 5x = =? 13 5x 5 = 20 5 x = 4 Divide both sides by 5. Solve each equation. Check your answers. 1. 3x 8 = 4 2. b 2 4 = y + 4 2y = = 3(x 2)

12 x-x 1-x 1-4 Solving Two-Step and Multi-Step Equations continued A two-step equation with fractions can be simplified by multiplying each side by the LCD. This will clear the fractions. Solve x = 2. Check: x = 2 x = 2 x = (12)2 Multiply both sides by the LCD x = 2 12 x = 12(2) 4 3 3x + 8 = 24 x is multiplied by 3. 8 is added. 8 8 Add 8 to both sides. 3x = 16 3x 3 = 16 3 x = =? =? =? =? 2 Divide both sides by 3. 2 =? 2 Solve each equation. Check your answers. 5. x = 1 6. w = = a

13 x-x 1-x 1-5 Solving Equations with Variables on Both Sides Variables must be collected on the same side of the equation before the equation can be solved. Solve 10x = 2x 16. Check: 10x = 2x 16 10x = 2x 16 2x 2x Add 2x to both sides. 10( 2) =? 2( 2) 16 8x = =? x 8 = 16 8 x = 2 Solve 3x = 5(x + 2). Divide both sides by =? 20 Check: 3x = 5x + 10 Distribute. 3x = 5(x + 2) 5x 5x Add 5x to both sides. 3( 5) =? 5( 5 + 2) 2x = =? 5( 3) 2x 2 = 10 2 x = 5 Divide both sides by =? 15 Write the first step you would take to solve each equation. 1. 3x + 2 = 7x 2. 4x 6 = 10x 3. 15x + 7 = 3x Solve each equation. Check your answers. 4. 4x + 2 = 5(x + 10) y + 3 = 4y (t + 7) + 2 = 6t 2 + 2t 1-38

14 x-x 1-x 1-5 Solving Equations with Variables on Both Sides continued Some equations have infinitely many solutions. These equations are true for all values of the variable. Some equations have no solutions. There is no value of the variable that will make the equation true. Solve 3x + 9 = 4x + 9 7x. Check any value of x: 3x + 9 = 3x + 9 Combine like terms. Try x = 4. 3x +3x Add 3x to each side. 3x + 9 = 4x + 9 7x 9 = 9 True statement. 3(4) + 9 =? 4(4) + 9 7(4) The solution is the set of all real numbers =? =? 3 Solve 2x x = 5x 10. Check any value of x: 2x x = 5x 10 Try x = 1. 5x + 6 = 5x 10 Combine like terms. 2x x = 5x 10 5x 5x Add 5x to each side. 2(1) (1) =? 5(1) 10 6 = 10 False statement =? 5 10 There is no solution. 11 =? 5 Solve each equation. 7. x + 2 = x x + 8 = 2x g = 3g x 1 4x = x (f + 3) + 4f = 6 + 6f 12. 3x + 7 2x = 4x

15 x-x 1-6 Solving for a Variable Solving for a variable in a formula can make it easier to use that formula. The process is similar to that of solving multi-step equations. Find the operations being performed on the variable you are solving for, and then use inverse operations. Operations Solve using Inverse Operations A = lw Solve for w. w is multiplied by l. Divide both sides by l. P = 2l + 2w Solve for w. w is multiplied by 2. Then 2l is added. Add 2l to both sides. Then divide both sides by 2. The formula A = 1 bh relates the area A of a triangle 2 to its base b and height h. Solve the formula for b. A = 1 2 bh b is multiplied by 1 2. The order of the inverse operations is the order of operations in reverse. 2 2 A = bh Multiply both sides by A = bh b is multiplied by h. 2A h = bh h 2A h = b Simplify. Divide both sides by h. Solve for the indicated variable. 1. P = 4s for s 2. a + b + c = 180 for b 3. P = KT V for K The formula V = 1 lwh relates the volume of a square pyramid 3 to its base length l, base width w, and height h. 4. Solve the formula for w. 5. A square pyramid has a volume of 560 in 3, a base length of 10 in., and a height of 14 in. What is its base width? 1-46

16 x-x 1-6 Solving for a Variable continued Any equation with two or more variables can be solved for any given variable. Solve x = y z 10 x = y z 10 for y. y z is divided by (x) = 10 y z Multiply both sides by x = y z +z +z 10x + z = y z is subtracted from y. Add z to both sides. Solve a = b + c d for c. a = b + c d b b Add b to each side. a b = c d c d(a b) = d d Multiply both sides by d. d(a b) = c Simplify. State the first inverse operation to perform when solving for the indicated variable. 6. y = x + z; for z 7. f + g 2 = h; for g 8. t = 3r + s ; for s 5 Solve for the indicated variable. 9. 3ab = c; for a 10. y = x + z m + 3 ; for z n = p; for m 1-47

17 x-x 1-7 Solving Absolute-Value Equations There are three steps in solving an absolute-value equation. First use inverse operations to isolate the absolute-value expression. Then rewrite the equation as two cases that do not involve absolute values. Finally, solve these new equations. Solve x = 8. Step 1: Isolate the absolute-value expression. x = Subtract 4 from both sides. x 3 = 4 Step 2: Rewrite the equation as two cases. x 3 = 4 Case 1 Case 2 Step 3: x 3 = 4 x 3 = 4 Solve Add 3 to both sides. x = 1 x = 7 The solution are 1 and 7. Solve each equation. 1. x 2 3 = 5 2. x = x 5 = x + 1 =

18 x-x 1-7 Solving Absolute-Value Equations continued Some absolute-value equations have two solutions. Others have one solution or no solution. To decide how many solutions there are, first isolate the absolute-value expression. Original Equation Simplified Equation Solutions x + 5 = 7 x + 5 = x = 2 x = 2 x = x 5 = 0 x = 1 x = x + 7 = 3 x = 2 has two solutions, x = 2 and x = 2. The solutions are 2 and 2. x 5 = 0 means x 5 = 0, so there is one solution x = 5. The solution is 5. x + 7 = 3 has no solutions because an absolute-value expression is never negative. There is no solution. Solve 2x = 7. 2x = Add 3 to both sides. 2x + 1 = 4 Absolute value cannot be negative. The equation has no solution. Solve each equation x 2 = 8 6. x = x 3 = x + 10 =

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