Unit 2: Accentuate the Negative Name: 1.1 Using Positive & Negative Numbers Number Sentence A mathematical statement that gives the relationship between two expressions that are composed of numbers and operation signs. For example, + 2 = 5 and 6 2 > 10 are number sentences; + 2, 5, 6 2, and 10 are expressions. Make a number sentence that would result in the following answers: a) 10 Ex: + 7 = 10, 1 = 10, 2 + 14 7 20 = 10 b) -5 Ex: 0 5 = 5, + 2 = 5 c) - Ex: 0 =, + 5 16 =
1.2 Number Line with Negatives Graphing Inequalities To graph an inequality: Put a closed dot for or or an open dot for > or <. Shade right for greater than or left for less than. a) x > 4 c) a > 0 b) b < -2 d) y < -1 Solving One Step Inequalities Opposites Solve and graph each inequality. a) x > 5 + + x > 7 9 b) c + 5 < -12-5 -5-1 - 17-16 c < -17 Two numbers whose sum is 0. For example, and are opposites. On a number line, opposites are the same distance from 0 but in different directions from 0. The number 0 is its own opposite. 1. Number Line Operations with Integers
Add integers using a number line: a) 2 + 4 = 6-1 0 1 2 4 5 6 7 b) -5 + - = - RULES FOR ADDING ON NUMBER LINE Always start at 0. For any positive number move to the right. For any negative number move to the left. - -7-6 -5-4 - -2-1 0 c) 5 + (-4) = 1-4 - -2-1 0 1 2 4 5 2.1 Adding Rational Numbers + 10 = 1 - + -10 = -1 Add: Two numbers with the same sign a) -12 + 19 = 7 b) 1 2 + 5 Add the absolute values and keep the common sign. c) 2. + (-.5) = -1.2 = 5 + 6 = 11 1 = -1 10 10 10 10 d) -2.1 + -. + 2.1 =. Two numbers with the different sign Subtract absolute values and keep the sign of the integer with greater absolute value. 4 + -7 = - -4 + 7 = Evaluate expressions: a) x + y for x = 1 1 + 5 = 4 = 11 and y = 5 b) a + b for a = -4.1 and b = -2.7-4.1 + -2.7 = -6.
2.2 Subtracting Rational Numbers Subtracting a number is the same as adding its opposite. Subtract: a) 6 = 6 + - (subtracting a positive is the same as adding a negative) = -2 b) -5 9 = -5 + -9 (subtracting a positive is the same as adding a negative) = -14 c) 5 ( 4 ) = 5 + 4 (subtracting a negative is the same as adding a positive) = 5 + 6 = 11 = 1 d) -1.2 (-1.) (subtracting a negative is the same as adding a positive) = -1.2 + 1. = 0.1 Evaluate a b for the given values. a) a = - and b = 2 b) a = 4.2 and b = -6.5-2 = - + -2 = -5 4.2 (-6.5) = 4.2 + 6.5 = 10.7 Additive Inverse Two numbers, a and b, that satisfy the equation a+b=0. For example, and are additive inverses, and 12 and 12 are additive inverses. Additive Identity Zero is the additive identity for rational numbers. Adding zero to any rational number results in a sum identical to the original rational number. For any rational number a, 0+a=a. For example, 0+4.75=4.75. Absolute Value The absolute value of a number is its distance from 0 on a number line. Numbers that are the same distance from 0 have the same absolute value. For example, and both have an absolute value of. 7 = 7 Absolute value of any number -7 = 7 is always positive. p Find the absolute value: a) 10 = 10 b) -5 = 5
.2/. Multiplying & Dividing Rational Numbers Rules for Multiplying and Dividing Two numbers with the same sign Positive answer 4 10 = 40-4 -10 = 40 Two numbers with the different sign Negative answer 5-7 = -5-5 7 = -5 LOVE and HATE If you love to love, then you If you love to hate, then you If you hate to love, then you If you hate to hate, then you Multiply or divide: a) -7 2 = -14 b) 40 - = -5 c) (4 5 ) = 12 40 d) 2.5(-4) = -10 = 10 e) - 2 5 (-1 1 ) = 17 5 ( 4 ) = 6 15 = 4 15. Long Division Terminating Decimals: A decimal that ends, or terminates, such as 0.5 or 0.125. Terminating decimals are rational numbers. Repeating Decimals: A decimal with a pattern of a fixed number of digits that repeats forever, such as 0.... and 0.7777....Repeating decimals are rational numbers. Use long division to write each fraction as a decimal. State whether each fraction will terminate or repeat. a) b) 5 6
4.1 Order of Operations -² = -( ) = -9 (-)² = - - = 9 Use order of operations to solve. Show all steps: a) 4 + (1 ) b) 24-6 + (-1 10) 4 + (1 + - + -) 24-6 + (-1 + -10) 4 + (-2 + -) 24-6 + -11 4 + -5-4 + -11-1 -15 c) -4 [7 (10 )²] d) (-42 6 + 4)² 2-4 [7 (2)²] (-7 + 4)² 2-4 [7 4] (-)² 2-4 9 2-4 + - = -7 1
4.2 Distributive Property Distributive Property: A mathematical property used to rewrite expressions involving addition and multiplication. a(b+c) = ab + ac. If an expression is written as a factor multiplied by a sum, you can use the Distributive Property to multiply the factor by each term in the sum. Calculated using Order of Operations: Calculated using Distributive Property: 5 ( 4 + 2) 5 ( 4 + 2) 5 (6) 5 4 + 5 2 0 20 + 10 0 Example using Distributive Property with variables: a) 4(5+x) = 4(5) + 4(x) = 20 + 4x b) (x + 7) = (x) + (7) = x + 21 c) x( 5) = x() x(5) = x 5x d) 2x( (-4)) = 2x( + 4) = 2x() + 2x(4) = 6x + x Using the distributive property to take out a common factor: a) 5x + 2x = x(5+2) b) -2x 7x = x(-2 7) c) -4(2) (-4)(5) = -4(2 5) Area model for distributive property: Write equivalent expressions to show two different ways to find the area of each rectangle x + x = 24 + x ( + x)