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Find the sum of -15 + (-46) + (-29)

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To do this, let's just first visualize

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what each of these numbers look like.

4
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So I'm going to draw a number line for each of them.

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-15 might look something like this.

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If this is 0, and let's say this is -15.

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I can represent -15 as an arrow that points from 0 to -15.

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The length of the arrow is the absolute value of this.

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It's the distance from 0.

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So the length here is 15

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and the negative says we are pointing to the left.

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So the absolute value is 15.

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That's the length of that arrow.

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Let's do the same thing for -46.

15
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Once again let me draw my number line.

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Put 0 is going to be right over there.

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and -46 is going to be someplace over here.

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Notice the same exact idea

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The distance between -46 and 0

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or another way to think that is the absolute value of -46.

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This distance right over here is going to be 46.

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And its direction is to the left.

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That's why we get to the number -46.

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The negative really tells you

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wether you're to the left or the right of 0.

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The absolute value says

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how far you are to the left or the right of 0.

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Finally let me do the same thing for -29.

29
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So once again, let me draw my number line.

30
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Use the yellow again.

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So got to draw my number line.

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Say this is 0, -29 is going to be roughly over here.

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Once again -29 is exactly 29 away from 0.

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So this length right over here is 29 and it is to the left

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that is why it is -29.

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If it was +29, it would be 29 to the right of 0.

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So we have represented all of these numbers

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and you can see what their absolute values are like.

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And now let's think what happens if you add them.

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One way to think about adding these numbers

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is as if you added these arrows.

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If you put this arrow on top of this arrow

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or to the left of this arrow

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If you started with this arrow leaves off,

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then you put this green arrow.

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And then you put the orange arrow.

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So let's do that. Let's draw that.

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It's going to be a longer arrow now.

49
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So we are going to start at 0.

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First we have -15.

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So we are going to move 15 spaces to the left,

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to get us to -15.

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Then we are going to go 46 spaces to the left,

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to get us to -15 + (-46)

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Let me draw that. Then we are going to go...

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We are going to figure out what that number is in a second.

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Then we're going to go 46 to the left. That's about that far.

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It's really just this arrow. I am now placing it.

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I am starting off where the -15 started

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or where the -15 left off.

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and then I am putting that arrow after that.

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And we don't know yet what number this gets us to.

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We are going to have to do a little math here.

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That's actually the point of this problem.

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But we know the length of this arrow is 46.

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It is 46 to the left.

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We know the length of this magenta arrow is 15.

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Then finally we have this orange arrow

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which we know has a length of 29,

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although it's 29 to the left.

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It looks like that.

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It has a length of 29.

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So when all is said and done, where are we?

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Where do we end up on our number line?

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Well, the total length that we are to the left,

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is going to be 15+46+29.

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But it's going to be to the left.

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So it's going to be negative.

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So we can really view this as

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Since all of these have the same sign

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Look this is the same thing

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as abs(-15)+abs(-46)+abs(-29)

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But then take the negative value of it.

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Let's just add that.

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15+46+29 is equal to...let's see.

86
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5+6 is 11. 11+9 is 20. Carry the 2.

87
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2+1 is 3. 3+4 is 7. 7+2 is 9. So you get 90.

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So this entire length here is 90.

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If you add up the arrows, you get a 90.

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But it is not 90 to the right.

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If it was 90 to the right, this would all be positive.

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Then we'll just get positive 90.

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But this is 90 to the left.

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So when you add these guys up,

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you don't get to positive 90.

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You get to negative 90.

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So one way to think about it is, these are all the same sign.

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So this is going to be equal to

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the -( abs(-15)+abs(-46)+abs(-29))

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and the reason why we are writing this.

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this might look fancy

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but this is really just the length of this purple arrow.

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This absolute value of 46,

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that's really just the length of the green arrow, 46.

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This is just the length of the orange arrow.

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So this is 15+46+29, gives us 90.

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But it's to the left. So that is why it is -90.