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Now let’s see what happens when the signs are different.
Model: $\mathrm{-5}+3.$
Interpret the expression. | $\mathrm{-5}+3$ means the sum of $\mathrm{-5}$ and $3$ . |
Model the first number. Start with 5 negatives. | |
Model the second number. Add 3 positives. | |
Remove any neutral pairs. | |
Count the result. | |
The sum of −5 and 3 is −2. | $\mathrm{-5}+3=\mathrm{-2}$ |
Notice that there were more negatives than positives, so the result is negative.
Model the expression, and then simplify:
$2+\left(\mathrm{-4}\right)$
2
Model the expression, and then simplify:
$2+(\mathrm{-5})$
−3
Model: $5+\left(\mathrm{-3}\right).$
Interpret the expression. | $5+\left(\mathrm{-3}\right)$ means the sum of $5$ and $\mathrm{-3}$ . |
Model the first number. Start with 5 positives. | |
Model the second number. Add 3 negatives. | |
Remove any neutral pairs. | |
Count the result. | |
The sum of 5 and −3 is 2. | $5+\left(\mathrm{-3}\right)=2$ |
Model the expression, and then simplify:
$\left(\mathrm{-2}\right)+4$
−2
Model each addition.
ⓐ | |
$4+2$ | |
Start with 4 positives. | |
Add two positives. | |
How many do you have? | $6\phantom{\rule{0.6em}{0ex}}$ $4+2=6$ |
ⓑ | |
$-3+6$ | |
Start with 3 negatives. | |
Add 6 positives. | |
Remove neutral pairs. | |
How many are left? | |
$3\phantom{\rule{0.6em}{0ex}}\mathrm{-3}+6=3$ |
ⓒ | |
$4+(\mathrm{-5})$ | |
Start with 4 negatives. | |
Add 5 negatives. | |
Remove neutral pairs. | |
How many are left? | |
$\mathrm{-1}\phantom{\rule{0.6em}{0ex}}4+(\mathrm{-5})=\mathrm{-1}$ |
ⓓ | |
$\mathrm{-2}+(\mathrm{-3})$ | |
Start with 2 negatives. | |
Add 3 negatives. | |
How many do you have? | $\mathrm{-5}\phantom{\rule{0.6em}{0ex}}\mathrm{-2}+(\mathrm{-3})=\mathrm{-5}$ |
Model each addition.
Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers.
For example, if you want to add $37+\left(\mathrm{-53}\right),$ you don’t have to count out $37$ blue counters and $53$ red counters.
Picture $37$ blue counters with $53$ red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because $53\mathrm{-37}=16,$ there are $16$ more negative counters.
Let’s try another one. We’ll add $\mathrm{-74}+\left(\mathrm{-27}\right).$ Imagine $74$ red counters and $27$ more red counters, so we have $101$ red counters all together. This means the sum is $\text{\u2212101.}$
$5+3$ | $\mathrm{-5}+\left(\mathrm{-3}\right)$ |
both positive, sum positive | both negative, sum negative |
When the signs are the same, the counters would be all the same color, so add them. | |
$\mathrm{-5}+3$ | $5+\left(\mathrm{-3}\right)$ |
different signs, more negatives | different signs, more positives |
Sum negative | sum positive |
When the signs are different, some counters would make neutral pairs; subtract to see how many are left. |
Simplify:
ⓐ Since the signs are different, we subtract $19$ from $47.$ The answer will be negative because there are more negatives than positives.
ⓑ The signs are different so we subtract $32$ from $40.$ The answer will be positive because there are more positives than negatives
Simplify each expression:
Simplify each expression:
Simplify: $\mathrm{-14}+(\mathrm{-36}).$
Since the signs are the same, we add. The answer will be negative because there are only negatives.
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