The addition and subtraction of real numbers are based on certain key principles. All principles will be explained using two real numbers, but certainly, the numbers in the exercise do not influence the method of resolution, therefore, these principles can be applied to any number in the exercise.

• When we have two real numbers with the same sign (plus or minus), this sign will remain in the result, which will, in fact, be the result of the addition. That is, if both numbers have a plus sign the result of the addition will also be positive. If both numbers have a minus sign the result of the subtraction will also be negative.
$+6+4=+10$
$-6-4=-10$

• When we have two numbers with different signs it is crucial to determine which of the two has the greater absolute value (absolute: the distance from zero). The larger number will determine the sign of the result and, in fact, we will perform a subtraction operation.
$+6-4=+2$
$-6+4=-2$

• When we have an exercise with a sequence of two signs (usually separated by parentheses) we will differentiate between several cases:

• When the sequence is of two plus signs the result will also be positive
$6+(+4)=+10$

• When the sequence is of two minus signs the result will also be positive
$6-(-4)=+10$

• When the sequence is of minus and plus or of plus and minus the result will be negative.
$6+(-4)=+2$
$6-(+4)=+2$

Examples with solutions for Addition and Subtraction of Directed Numbers

Exercise #1

$(+6)-(+11)=$

Step-by-Step Solution

Let's remember the rule:

$-(+x)=-x$

Now let's write the exercise in the appropriate form:

$6-11=$

We'll locate the number 6 on the number line and from there we'll move 11 steps to the left:

$-5$

Exercise #2

$14-(-3)=$

Step-by-Step Solution

Let's remember the rule:

$-(-x)=+$

We'll write the exercise in the appropriate form:

$14+(3)=$

We'll locate the number 14 on the number line, from which we'll move 3 steps to the right (since 3 is greater than zero):

We can see that we've reached the number 17.

$17$

Exercise #3

$3+(-4)=$

Step-by-Step Solution

We will locate the number 3 on the number line, then move 4 steps to the left from it (since minus 4 is less than zero):

We can see that we have reached the number minus 1.

$-1$

Exercise #4

$(+8)+(+12)=$

Step-by-Step Solution

Let's place 8 on the number line and move 12 steps to the right.

Let's note that our result is a positive number:

.

Let's solve the exercise:

$8+12=20$

$20$

Exercise #5

$(-8)-(-13)=$

Step-by-Step Solution

Let's remember the rule:

$-(-x)=+x$

Now let's write the exercise in the appropriate form:

$-8+13=$

We'll use the substitution law and solve:

$13-8=5$

$5$

Exercise #6

$(-10)-(+13)=$

Step-by-Step Solution

Let's locate -10 on the number line and move 13 steps to the left.

Let's note that our result is a negative number:

Let's remember the rule:

$-(+x)=-x$

Now let's write the exercise in the appropriate form and solve it:

$-10-13=-23$

$-23$

Exercise #7

$-5-(-2)=$

Step-by-Step Solution

Let's remember the rule:

$-(-x)=+x$

Therefore, the exercise we received is:

$-5+2=$

We'll locate minus 5 on the number line and move two steps to the right (since 2 is greater than zero):

We can see that we've arrived at the number minus 3.

$-3$

Exercise #8

$3-(-2)=$

Step-by-Step Solution

Let's remember the rule:

$-(-x)=+$

We'll write the exercise in the appropriate form:

$3+(2)=$

We'll locate the number 3 on the number line, from which we'll move 2 steps to the right (since 2 is greater than zero):

We can see that we've reached the number 5.

$5$

Exercise #9

$(-8)+(-12)=$

Step-by-Step Solution

Let's locate -8 on the number line and move 12 steps to the left.

Let's note that our result is a negative number:

Let's remember the rule:

Now let's write the exercise in the appropriate form and solve it:

$-8-12=-20$

$-20$

Exercise #10

$(-8)+(+12)=$

Step-by-Step Solution

Let's remember the rule:

$+(+x)=+x$

Now let's write the exercise in the following way:

$-8+12=$

We'll draw a number line and place minus 8 on it, then move 12 steps to the right:

Therefore:

$-8+12=4$

$4$

Exercise #11

$-4+(-2)=$

Step-by-Step Solution

We'll locate minus 4 on the number line and move two steps to the left (since minus 2 is less than zero):

We can see that we've arrived at the number minus 6.

$-6$

Exercise #12

$-3-(-4)=$

Step-by-Step Solution

Let's remember the rule:

$-(-x)=+$

We'll write the exercise in the appropriate form:

$-3+(4)=$

We'll locate the number negative 3 on the number line, from which we'll move 4 steps to the right (since 4 is greater than zero):

We can see that we've reached the number 1.

$1$

Exercise #13

$12-?=15$

Step-by-Step Solution

We know that the distance between 12 and 15 is three steps.

In other words:

$12+3=15$

Now let's remember the rule:

$-(-x)=+$

We'll substitute x with the number 3 and get:

$12-(-3)=15$

Therefore, the answer is minus 3

$-3$

Exercise #14

$-25-?=100$

Step-by-Step Solution

We'll add ?+ to both sides to zero out the left side:

$-25-?+?=100+?$

$-?+?=0$

Therefore, we get:

$-25-0=100+\text{?}$

$-25=100+\text{?}$

We'll add minus 100 to both sides to zero out the right side:

$-25-100=100-100+\text{?}$

Now we get:

$-125=0+\text{?}$

$-125=\text{?}$

$-125$

Exercise #15

$12-(-2)=$

Step-by-Step Solution

Let's remember the rule:

$-(-x)=+$

We'll write the exercise in the appropriate form:

$12+(2)=$

We'll locate the number 12 on the number line, from which we'll move 2 steps to the right (since 2 is greater than zero):

We can see that we've reached the number 14.

$14$