§ 1.2 Operations with Real Numbers and Simplifying Algebraic Expressions

Blitzer, Algebra for College Students, 6e – Slide #2 Section 1.2 Finding Absolute Value Absolute value is used to describe how to operate with positive and negative numbers. The absolute value of -5 is 5 because -5 is 5 units from 0 on the number line. The absolute value of 3 is +3 because 3 is 3 units from 0 on the number line. Geometric Meaning of Absolute Value The absolute value of a real number a, denoted is the distance from 0 to a on the number line. This distance is always nonnegative.

Blitzer, Algebra for College Students, 6e – Slide #3 Section 1.2 Rules for Addition of Real Numbers To add two real numbers with like signs, add their absolute values. Use the common sign as the sign of the sum. To add two real numbers with different signs, subtract the smaller absolute value from the greater absolute value. Use the sign of the number with the greater absolute value as the sign of the sum.

Blitzer, Algebra for College Students, 6e – Slide #4 Section 1.2 Adding Real Numbers Add: -12+(-5) EXAMPLE We are adding numbers having like signs. So we just add the absolute values and take the common sign as the sign of the sum. Answer: -17 EXAMPLE Add: We are adding numbers having unlike signs. We just take the difference of the absolute values (difference is 4) and then take the sign of the number that has the largest absolute value (that’s the 14 and it is positive). Answer: +4

Blitzer, Algebra for College Students, 6e – Slide #5 Section 1.2 Subtracting Real Numbers Definition of Subtraction If a and b are real numbers, a – b = a + (-b) That is, to subtract a number, just add its additive opposite (called its additive inverse).

Blitzer, Algebra for College Students, 6e – Slide #6 Section 1.2 Subtracting Real Numbers Subtract: -12-(-5) EXAMPLE Here, change the subtraction to addition and replace -5 with its additive opposite. That is, replace the -(-5) with (-5) EXAMPLE Subtract: (-4) -14 Here, change the subtraction to addition and replace 4 with its additive opposite of -4. Then you use the rule for adding two negative numbers.

Blitzer, Algebra for College Students, 6e – Slide #7 Section 1.2 Multiplying Real Numbers RuleExamples An odd number of negatives makes the answer negative. (-4)8 = -32 (-3)(-10)(-6) = -180 An even number of negatives makes the answer positive. (-2)(-11) = (-8)5 = 160

Blitzer, Algebra for College Students, 6e – Slide #8 Section 1.2 Dividing Real Numbers Rules for Dividing Real Numbers The quotient of two numbers with different signs is negative. The quotient of two numbers with the same sign is positive. In either multiplication or division of signed numbers, it is important to count the negatives in the product or quotient: Odd number of negatives and the answer is negative. Even number of negatives and the answer is positive.

Blitzer, Algebra for College Students, 6e – Slide #9 Section 1.2 Order of OperationsEXAMPLE Simplify. SOLUTION Evaluating exponent Multiply Divide Subtract

Blitzer, Algebra for College Students, 6e – Slide #10 Section 1.2 Basic Algebraic Properties PropertyExamples Commutative = (3) = 3(2) = (10) = 10(4) = (8) = 8(7) Associative 4 + (3 + 2) = (4 + 3) + 2 (6 4)11 = 6(4 11) 3(2 5) = (3 2)5 Distributive 7(2x + 3) = 14x (3x-2-4y) = 15x – 10 – 20y (2x + 7)4 = 8x + 28