§ 1.2 Operations with Real Numbers and Simplifying Algebraic Expressions

Finding Absolute Value Absolute value is used to describe how to operate with positive and negative numbers. 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. 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. Blitzer, Algebra for College Students, 6e – Slide #2 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 #3 Section 1.2

Adding Real Numbers Add: -12+(-5) EXAMPLE Add: -12+(-5) 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 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). Add: -10 +14 Answer: +4 Blitzer, Algebra for College Students, 6e – Slide #4 Section 1.2

Adding Real Numbers Add: Using the rule, rewrite with absolute values. EXAMPLE Add: The two numbers in this example have different signs. We know that 2/5 > 3/20. We need to subtract the smaller absolute value from the larger and take the sign of the number having the greater absolute value. Our answer will be negative since the sign of 2/5 is negative. SOLUTION Using the rule, rewrite with absolute values. Then simplify. Blitzer, Algebra for College Students, 6e – Slide #5 Section 1.2

Adding Real Numbers Common denominators Multiply Subtract CONTINUED Common denominators Multiply Subtract Finally, simplify the fraction. Whew! This last example was a little difficult. In practice, we don’t always rewrite using the absolute values. We just learn the rules and carry out the computation without putting in all the formal steps. Blitzer, Algebra for College Students, 6e – Slide #6 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 #7 Section 1.2

Subtracting Real Numbers EXAMPLE Subtract: -12-(-5) Here, change the subtraction to addition and replace -5 with its additive opposite. That is, replace the -(-5) with 5. -12-(-5) -12+5 -7 EXAMPLE Subtract: -10 - (+4) 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. -10 +(-4) -14 Blitzer, Algebra for College Students, 6e – Slide #8 Section 1.2

Multiplying Real Numbers Rule Examples The product of two real numbers with different signs is found by multiplying their absolute values. The product is negative. (-4)8 = -32 The product of two real numbers with the same sign is found by multiplying their absolute values. The product is positive. (-2)(-11) = -22 The product of 0 and any real number is 0 0(-14) = 0 If no number is 0, a product with an odd number of negative factors is found by multiplying absolute values. The product is negative. (-3)(-10)(-6) = -180 If no number is 0, a product with an even number of negative factors is found by multiplying absolute values. The product is positive. -4(-8)5 = 160 Blitzer, Algebra for College Students, 6e – Slide #9 Section 1.2

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 #10 Section 1.2

Dividing Real Numbers Divide. EXAMPLE SOLUTION Blitzer, Algebra for College Students, 6e – Slide #11 Section 1.2

Order of Operations Simplify. Evaluating exponent Multiply Divide EXAMPLE Simplify. SOLUTION Evaluating exponent Multiply Divide Subtract Blitzer, Algebra for College Students, 6e – Slide #12 Section 1.2

Basic Algebraic Properties Property Examples Commutative 2 + 3 = 3 + 2 2(3) = 3(2) 10 + 4 = 4 + 10 4(10) = 10(4) 8 + 7 = 7 + 8 7(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 + 21 5(3x-2-4y) = 15x – 10 – 20y (2x + 7)4 = 8x + 28 Blitzer, Algebra for College Students, 6e – Slide #13 Section 1.2

Combining Like Terms Simplify: 3a – (2a + 4b – 6c) +2b – 3c EXAMPLE Simplify: 3a – (2a + 4b – 6c) +2b – 3c SOLUTION 3a – (2a + 4b – 6c) +2b – 3c Distributive Property 3a – 2a - 4b + 6c +2b – 3c (3a – 2a) + (2b - 4b) + (6c – 3c) Comm. & Assoc. Prop. (3 – 2)a + (2 - 4)b + (6 – 3)c Distributive Property 1a - 2b + 3c Subtract Simplify a - 2b + 3c Blitzer, Algebra for College Students, 6e – Slide #14 Section 1.2