2Absolute Value-7+7-5510-10A number line has many functions. 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.
3Absolute Value-7+7-5510-10Both of these numbers, positive seven (+7) and negative seven (-7), represent a point that is seven units away from the origin.The absolute value of a number is the distance between that number and zero on a number line. Absolute value is shown by placing two vertical bars around the number as follows:5 The absolute value of five is five.-3 The absolute value of negative three is three.
4Adding Integers - Same Sign What is 5 + 7?We can show how to do this by using algebra tiles.+=5+7=12What is –5 + -7?=+-12-5+-7=
5Adding Integers - Same Sign We can show this same idea using a number line.9-5510-10What is 5 + 4?Move five (5) units to the right from zero.Now move four more units to the right.The final point is at 9 on the number line.Therefore, = 9.
6Adding Integers - Same Sign -9-5510-10What is -5 + (-4)?Move five (5) units to the left from zero.Now move four more units to the left.The final point is at -9 on the number line.Therefore, -5 + (-4) = -9.
7Adding Integers - Same Sign To add integers with the same sign, add the absolute values of the integers. Give the answer the same sign as the integers.ExamplesSolution
8Additive Inverse What is (-7) + 7? -5 5 10 -10 10-10To show this, we can begin at zero and move seven units to the left.Now, move seven units to the right.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
9Additive InverseWhen using algebra tiles, the additive inverses make what is called a zero pair.For example, the following is a zero pair.1 + (-1) = 0.This also represents a zero pair.x + (-x) = 0
10Adding Integers - Different Signs Add the following integers: (-4) + 7.-5510-10Start at zero and move four units to the left.Now move seven units to the right.The final position is at 3.Therefore, (-4) + 7 = 3.
11Adding Integers - Different Signs Add the following integers: (-4) + 7.-5510-10Notice that seven minus four also equals three.In our example, the number with the larger absolute value was positive and our solution was positive.Let’s try another one.
12Adding Integers - Different Signs -5510-10Start at zero and move nine places to the left.Now move three places to the right.The final position is at negative six, (-6).Therefore, (-9) + 3 = -6.
13Adding Integers - Different Signs -5510-10In this example, the number with the larger absolute value is negative. The number with the smaller absolute value is positive.We know that 9-3 = 6. However, (-9) + 3 = -6.6 and –6 are opposites. Comparing these two examples shows us that the answer will have the same sign as the number with the larger absolute value.
14Adding Integers - Different Signs 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.ExampleSubtractSolution
15Subtracting IntegersWe can show the addition of numbers with opposite signs by using algebra tiles. For example, 3 + (-5) would look like this:+Group the zero pairs.Remove the zero pairs
16Subtracting IntegersWe can show the addition of numbers with opposite signs by using algebra tiles. For example, 3 + (-5) would look like this:+=Group the zero pairs.Remove the zero pairsThe remainder is the solution.Therefore, 3 + (-5) = -2
17Subtracting IntegersLet’s see how subtraction works using algebra tiles forBegin with three algebra tiles.Now, take away or remove five tiles from these three. It can’t be done. However, we can add zero pairs until we have five ones because adding zero doesn’t change the value of the number.
18Subtracting IntegersLet’s see how subtraction works using algebra tiles forBegin with three algebra tiles.Now remove the five ones.
19Subtracting IntegersLet’s see how subtraction works using algebra tiles forBegin with three algebra tiles.Now remove the five ones.The remainder is negative two (-2).Therefore, 3 – 5 = -2. This is the same as 3 + (-5).To subtract a number, add its additive inverse.For any numbers a and b, a – b = a + (-b).
20You Try It! Find each sum or difference. -24 – (-40)– (-14) –Simplify each expression.r – 27r c – (-12c) x + 45xEvaluate each expression.89 -14 -7 c + 5 if c = -19