Notes 7th Grade Math McDowell Chapter 2

ONE STEP EQUATIONS 9/25 equations An equation is like a balance scale because it shows that two quantities are equal. What you do to one side of the equation must also be done to the other side to keep it balanced.

Addition property of equality Add the same thing on both sides to keep the equation equal. Subtraction property of equality Subtract the same thing on both sides to keep the equation equal.

Ask three questions about the equation: solving one step equations Ask three questions about the equation: 1. What is the variable? 2. What operation is performed on the variable? 3. What is the inverse operation? (The one that will undo what is being done to the variable)

Solve x + 4 = 12 Example x Addition. Subtraction. x + 4 = 12 - 4 - 4 x 1. What is the variable? 2. What operation is being performed on the variable? Addition. 3. What is the inverse operation (the one that will undo what is being done to the variable)? Subtraction. The subtraction property of equality tells us to subtract the same thing on both sides to keep the equation equal. x + 4 = 12 - 4 - 4 x = 8

Solve y - 7 = -13 Example y Subtraction. Addition. y - 7 = -13 + 7 + 7 1. What is the variable? 2. What operation is being performed on the variable? Subtraction. 3. What is the inverse operation (the one that will undo what is being done to the variable)? Addition. The addition property of equality tells us to add the same thing on both sides to keep the equation equal. y - 7 = -13 + 7 + 7 y = -6

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Multiply the same thing on both sides to keep the equation equal. 9/28 Multiplication property of equality Multiply the same thing on both sides to keep the equation equal. Division property of equality Divide the same thing on both sides to keep the equation equal.

Solve –6a = 12 Example a Multiplication. Division –6a = 12 -6 -6 a = 1. What is the variable? 2. What operation is being performed on the variable? Multiplication. 3. What is the inverse operation (the one that will undo what is being done to the variable)? Division The division property of equality tells us to divide the same thing on both sides to keep the equation equal. –6a = 12 -6 -6 a = -2

-10 Solve = b Division. Multiplication = -10 2 • • 2 b = -20 1. What is the variable? 2. What operation is being performed on the variable? Division. 3. What is the inverse operation (the one that will undo what is being done to the variable)? Multiplication The multiplication property of equality tells us to multiply the same thing on both sides to keep the equation equal. = -10 2 • = -10 • 2 b = -20

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Evaluating and Writing Algebraic Expressions 9/29 Variable Letters that stand for numbers x + 4 = 5, x is 1 Variables are used when we don’t know the number

Expression A mathematical phrase No = sign 3 + 7 Algebraic Expression A mathematical phrase that uses variables, numerals, and operation symbols 4a + b a and b are both variables

Words that mean math Addition Subtraction increased by more than combined, together total of sum added to Subtraction decreased by minus, less difference between/of less than, fewer than

Words that mean math Multiplication Division Of, times, multiplied by, product of, increased/decreased by a factor of Division per, a, out of, ratio of, quotient of, percent (divide by 100)

Words that mean math Equals is, are, was, were, will be gives, yields sold for

Examples 12 inches per foot 12 in / ft 2. 10 packs of x crayons 10x 3. 7 less than t t – 7 (this one is written backwards from the word order—be careful)

Reminder Addition and multiplication are commutative Order is not a issue Subtraction and division are not Order matters A LOT! Reminder

You Try The value in cents of q quarters 13° warmer than the room temperature t The sum of 18 and b 5 less than a The quotient of d and 6

Evaluating Expressions 9/29 Evaluate an expression Simplify the expression by combining like terms Combine numbers with numbers and like variables with like variables Evaluate a variable expression 1. Plug in the number the variable is standing in for 2. Simplify by combining like terms

Plug Chug & Evaluate a variable expression in the number the variable is standing in for Plug & chug chug out the answer by combining like terms Chug

Example Plug Chug Evaluate 2x + x for x = 4 2(4) + 4 8 + 4 12 In the number 2(4) + 4 Chug Out the answer 8 + 4 12

Example Plug Chug Evaluate 2ab for a = 3, b = 4 2(3)(4) 6(4) 24 In the numbers 2(3)(4) Chug Out the answer 6(4) 24

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Evaluating Two-Step Equations 9/30 Two operations are required to isolate x Plug and Chug to check your answer

Ask five questions about the equation: solving two step equations 1. What is the variable? 2. What is the first operation being done to the variable? 3. What is the inverse operation? 4. What is the second operation being done to the variable? 5. What is the inverse operation?

Work Backwards First—undo any addition or subtraction Second—undo any multiplication or division

Example 2 · · 2 = - 4 x = -8 -4 – 3 = -7 -7 = -7 + 3 + 3 Draw “the road” Work Backwards + 3 + 3 Add 3 to both sides x  2 • 2 -3 +3 Simplify 2 · · 2 = - 4 Multiply both sides by 2 Simplify x = -8 Check your answer Plug and Chug -4 – 3 = -7 -7 = -7

Solve Remember to work backwards!!!! Additional Examples Addition or subtraction 1st!! Additional Examples Solve 1. 2x - 1 = -3 2. 3. 5z + 16 = 51

Video http://www.phschool.com/atschool/academy123/html/bbapplet_wl-problem-430737.html Interactive http://www.mathsnet.net/algebra/balance.html

You try Workbook p 32 # 1-29 odd

Graphing and Writing Inequalities 10/1 Inequality signs < > £ ³ Open dot Closed dot less than fewer than greater than more than exceeds less than or equal to no more than at most greater than or equal to no less than at least

Solution of An Inequality Any value that makes the inequality true There can be infinity many solutions Which are solutions of x  -3? 5, 9, -2, or –37 Replace x with the possible solution and determine if it is true.

≤ Has a closed circle and points left Graphing Inequalities Graph c > -2. > Has an open circle and points right Graph m ≤ 4. ≤ Has a closed circle and points left

Writing Inequalities from the graph Write an inequality from each graph Closed circle and points right  Circle is over the 1 x  1 Open circle and points left > Circle is over the -4 x > -4.

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Solving Inequalities by Adding or Subtracting 10/20 The Inequality sign When adding or subtracting, treat the inequality sign just like you would an = sign Perform the inverse to isolate the variable and leave the inequality sign alone.

Ask three questions about the inequality: Reminder Ask three questions about the inequality: 1. What is the variable? 2. What operation is performed on the variable? 3. What is the inverse operation? (The one that will undo what is being done to the variable)

Solve x + 7 > 15 Example x Addition. Subtraction. x + 7 > 12 - 7 1. What is the variable? 2. What operation is being performed on the variable? Addition. 3. What is the inverse operation (the one that will undo what is being done to the variable)? Subtraction. The subtraction property of inequality tells us to subtract the same thing on both sides to keep the equation equal. x + 7 > 12 - 7 - 7 x > 5

Example y Subtraction. Addition. y – 10  -12 +10 +10 y  -2 Solve y – 10  -12 Example y 1. What is the variable? 2. What operation is being performed on the variable? Subtraction. 3. What is the inverse operation (the one that will undo what is being done to the variable)? Addition. The addition property of inequality tells us to add the same thing on both sides to keep the equation equal. y – 10  -12 +10 +10 y  -2

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Solving Inequalities by Multiplying or Dividing 10/23 When multiplying or dividing on both sides of an inequality sign watch for negatives. The Inequality sign When multiplying or dividing by a positive number leave the inequality sign alone. When multiplying or dividing by a negative number, flip the inequality sign.

Solve 2a < 10 Example a Multiply by 2 Divide by 2 2a < 10 2 2 1. What is the variable? 2. What operation is being performed on the variable? Multiply by 2 3. What is the inverse operation (the one that will undo what is being done to the variable)? Divide by 2 The division property of inequality tells us to divide the same thing on both sides to keep the equation equal. 2a < 10 2 is positive so leave the inequality sign alone. 2 2 a < 5

Solve m/-4  3 Example m Divide by -4 Multiply by -4 m/-4  3 X-4 X-4 1. What is the variable? 2. What operation is being performed on the variable? Divide by -4 3. What is the inverse operation (the one that will undo what is being done to the variable)? Multiply by -4 The division property of inequality tells us to divide the same thing on both sides to keep the equation equal. m/-4  3 -4 is negative so flip the inequality sign. X-4 X-4 m  -12

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Evaluating Two-Step Inequalities 10/27 Two operations are required to isolate x Plug and Chug to check your answer Remember: When multiplying or dividing by a negative number, flip the inequality sign.

Ask five questions about the inequality: Solving Two Step Inequalities 1. What is the variable? 2. What is the first operation being done to the variable? 3. What is the inverse operation? 4. What is the second operation being done to the variable? 5. What is the inverse operation?

Work Backwards First—undo any addition or subtraction Second—undo any multiplication or division

Example -2x – 5  25  30 x  -15 -2(-15) – 5  25 30 – 5  25 25  25 Draw “the road” Example -2x – 5  25 Work Backwards + 5 + 5 Add 5 to both sides Simplify -2x  30 x • -2  -2 -5 +5 -2 -2 Divide both sides by -2 Simplify x  -15 When multiplying or dividing by a negative number, flip the inequality sign. -2(-15) – 5  25 30 – 5  25 Check your answer Plug and Chug 25  25 Graph -17 -16 -15 -14 -13

You Try Simplify and graph y – 4 > -12 2 2. –4m + 10  30 3. ¼a – 13 < -3

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