Adding and Subtracting Signed Integers

The Number Line -7 +7 -5 5 10 -10 Previously, we learned that numbers to the right of zero are positive and numbers to the left of zero are negative. By putting points on the number line, we can graph values. If one were to start at zero and move seven places to the right, this would represent a value of positive seven. If one were to start at zero and move seven places to the left, this would represent a value of negative seven.

Adding Integers - Same Sign We can show addition using a number line. 9 -5 5 10 -10 What is 5 + 4? Start at five (5 units to the right from zero). Move four units to the right. The final point is at 9 on the number line. Therefore, 5 + 4 = 9.

Adding Integers - Same Sign Now add two negative numbers on a number line. -9 -5 5 10 -10 What is -5 + (-4)? Start at –5 (5 units to the left from zero). Move four units to the left. Go left since we are adding a negative number. The final point is at -9 on the number line. Therefore, -5 + (-4) = -9.

Adding Integers - Same Sign We can also show how to do this by using algebra tiles. Each dark tile is a positive 1, each light tile is a negative 1 What is 5 + 7? + = 5 + 7 = 12 What is –5 + –7? = + -12 -5 + -7 =

Adding Integers - Same Sign RULE: To add integers with the same sign, add the absolute values of the integers. Give the answer the same sign as the integers. Examples Solution

Additive Inverse What is (-7) + 7? -5 5 10 -10 10 -10 To show this, start at the value -7 (seven units left of zero). Now, move seven units to the right (adding positive seven). Notice, we are back at zero (0). For every positive integer on the number line, there is a corresponding negative integer. These integer pairs are opposites or additive inverses. Additive Inverse Property – For every number a, a + (-a) = 0.

Additive Inverse When using algebra tiles, the additive inverses make what is called a zero pair. For example, the following is a zero pair, the two tiles cancel each other out. 1 + (-1) = 0. This also represents a zero pair. x + (-x) = 0

Adding Integers - Different Signs Add the following integers: (-4) + 7. -5 5 10 -10 Start at the value -4 (four units to the left of zero). Move seven units to the right (because we are adding a positive number. The final position is at 3. Therefore, (-4) + 7 = 3.

Adding Integers - Different Signs -5 5 10 -10 Start at the value -9 (nine places to the left of zero). Move three places to the right (adding a positive number). The final position is at negative six, (-6). Therefore, (-9) + 3 = -6.

Adding Integers - Different Signs -5 5 10 -10 Start at the value 7 (seven places to the right of zero). Move three places to the left (adding a negative number). The final position is at positive four, (4). Therefore, 7 + (-3) = 4.

Adding Integers - Different Signs Each dark tile is a positive 1, each light tile is a negative 1. One + 1 and one - 1 make a zero pair, they cancel each other. What is (- 4) + 7? + = -4 + 7 = 3 What is 4 + (-7)? = + -3 4 + -7 =

Adding Integers - Different Signs RULE: To add integers with different signs determine the absolute value of the two numbers. Subtract the smaller absolute value from the larger absolute value. The solution will have the same sign as the number with the larger absolute value. Example Subtract Solution

Subtracting Integers Keep-Change-Change. 12 - 9 = 12 + (-9) = 3 Subtraction is defined as addition: a - b = a + (-b). To perform subtraction, remember this rule: Keep-Change-Change. 12 - 9 = 12 + (-9) Keep the 1st # the same. Change the sign of the 2nd #. Change the minus to a plus. = 3 4 - 13 = 4 + (-13) Keep the 1st # the same. Change the sign of the 2nd #. Change the minus to a plus. = -9

Subtracting Integers = 7 4 - (-3) = 4 + (3) = -2 -4 - (-2) = -4 + (2) Keep-Change-Change doesn’t mean the 2nd number always ends up being negative: 4 - (-3) = 4 + (3) Keep the 1st # the same. Change the sign of the 2nd #. Change the minus to a plus. = 7 -4 - (-2) = -4 + (2) Keep the 1st # the same. Change the sign of the 2nd #. Change the minus to a plus. = -2

You Try It! Find each sum or difference. 1. -24 – 11 2. 18 + (-40) 1. -24 – 11 2. 18 + (-40) 3. -9 + 9 4. -16 – (-14) 5. 13 – 35 6. -29 + 65 Simplify each expression. 7. 18r – 27r 8. 9c – (-12c) 9. -7x + 45x 10. -3y + (-7y)

Solutions 1. -24 – 11 = -24 + (-11) = -35 2. 18 + (-40) 3. -9 + 9 1. -24 – 11 2. 18 + (-40) 3. -9 + 9 4. -16 – (-14) 5. 13 – 35 6. -29 + 65 = -24 + (-11) = -35 = -22 = 0 = -16 + (14) = -2 = 13 + (-35) = -22 = 36

Solutions 7. 18r – 27r = 18r + (-27r) = -9r 8. 9c – (-12c) 9. -7x + 45x 10. -3y + (-7y) = 18r + (-27r) = -9r = 9c + (12c) = 21c = 38x = -10y