Chapter 02 Section 03 Adding and Subtracting Integers

Chapter 02 – Section 03 Adding and Subtracting Integers

To find the absolute value of a number and add and subtract integers. What is absolute value? • absolute value - distance between zero and a number on a number line Since distance can never be less than zero, absolute values are ALWAYS positive numbers. The symbol for the absolute value of a number is two vertical bars around the number. |-125| = 125 is read The absolute value of -125 equals 125. |200| = 200 is read The absolute value of 200 equals 200. |-7 + 6| = 1 is read The absolute value of the quantity -7 + 6 equals 1. © William James Calhoun

EXAMPLE 1α: Evaluate -|x + 6| if x = -10. Substitute -10 for x. Add inside the absolute value bars. -|-10 + 6| -|-4| Take the absolute value of -4. -(4) Drop the parenthesis. -4 EXAMPLE 1β: Evaluate each expression if r = -5 and p = 3. a. |r + 2| b. -|4 + r| c. -|7 + p| © William James Calhoun

2. 3. 1 ADDING INTEGERS To add integers with the same sign, add their absolute values. Give the result the same sign as the integers. To add integers with different signs, subtract the lesser absolute value from the greater absolute value. Give the result the same sign as the integer with the greater absolute value. To add integers with the same sign, add their absolute values. Give the result the same sign as the integers. If the signs are alike, add and keep the sign. To add integers with different signs, subtract the lesser absolute value from the greater absolute value. Give the result the same sign as the integer with the greater absolute value. If the signs are different, subtract and keep the sign of the “stronger” one. © William James Calhoun

EX 2β EXAMPLE 2α: Find each sum. a. -10 + (-17) METHOD 1: -10 + (-17) = -(|-10| + |-17|) = -(10 + 17) = -27 b. 39 + (-22) METHOD 1: 39 + (-22) = +(|39| - |-22|) = +(39 - 22) = 17 METHOD 2: Both numbers being added are The signs are different on the two negative. terms, so subtract the weights of Add the weight of each number and take the sign of keep the negative sign. the “stronger” number. 10 + 17 = 27 39 - 22 = 17 -27 is the answer. 17 is the answer. METHOD 2: c. -28 + 16 The signs are different on the two terms, so METHOD 1: subtract and take the sign of the “stronger” number. -28 + 16 = -(|-28| - |16|) 28 - 16 = 12 = -(28 - 16) -12 is the answer. = -12 © William James Calhoun

EXAMPLE 2β: Find each sum. a. -6 + (-2) b. -56 + 42 c. 38 + (-36) © William James Calhoun

More terms. • opposites - numbers with the same absolute value but different signs • additive inverses - two numbers that add to zero Think about it: The two number must be opposites to add to zero! 2. 3. 2 ADDITIVE INVERSE PROPERTY For every number a, a + (-a) = 0. 5 has an opposite, -5. -5 has an opposite, 5. Adding 5 + (-5) or -5 + 5 equals 0. © William James Calhoun

2. 3. 3 SUBTRACTING INTEGERS To subtract a number, add its additive inverse. For any numbers a and b, a - b = a + (-b). To subtract a number is to add its opposite. Taking away 3 is the same as adding -3. Subtracting numbers and adding negative numbers are one in the same. You have to train yourself in this VERY IMPORTANT skill! Example: 6 - 2 = 4 and 6 + (-2) = 4 Additive Inverses Same Result © William James Calhoun

EX 3β EXAMPLE 3α: Find each difference. a. 6 - 14 To subtract 14, add -14. 6 - 14 = 6 + (-14) signs opposite so subtract and take sign of stronger = -8 b. -12 - (-8) To subtract -8, add its inverse, +8. -12 - (-8) = -12 + 8 signs opposite so subtract and take sign of stronger = -4 c. 43 - (-26) = 43 + 26 = 69 Special note here: To subtract a negative is to do the opposite of subtraction – adding. So, minus-a-minus is a plus. © William James Calhoun

EXAMPLE 3β: Find each difference. a. -9 – (-4) b. -3 – 7 c.

EX 4β The rule for subtracting integers and the distributive property can be used to combine like terms. EXAMPLE 4α: Simplify 2 x - 5 x. 2 x – 5 x Use the additive inverse. = 2 x + (-5)x Use the distributive property. = [2 + (-5)]x Two plus negative five yields… = -3 x EXAMPLE 4β: Simplify 4 a – 9 a. © William James Calhoun

Solving the problems in this section can be done in many different ways. The methods presented show the actual step-by-grueling-step processes. When you feel comfortable doing so, steps should be skipped and quicker means employed. DO THESE PROBLEMS HOW YOU FEEL MOST COMFORTABLE - AS LONG AS YOUR WAY GIVES YOU A CORRECT ANSWER! © William James Calhoun