Addition and Subtraction of Directed Numbers: Solving the problem

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

Note that our result is a positive number:

Now solve the following exercise:

$8+12=20$

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:

$-8-12=-20$

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:

$-10-13=-23$

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:

$13-8=5$

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:

The answer is minus 5.

First, 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:

$-8+12=4$

Let's position minus $\frac{1}{7}$ on the number line and move one step to the right, since $\frac{7}{7}=\frac{1}{1}=1$

We should note that our result is a positive number:

Let's remember the rule:

$-(-x)=+x$

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

$-\frac{1}{7}+1=\frac{6}{7}$

First we need to locate the number 8 on the number line and move 4.5 steps to the left from it:

Remember the rule:

$+(-x)=-x$

Now let's rewrite the problem in the appropriate form and solve:

$8-4.5=3.5$

Let's remember the rule:

$-(-x)=+x$

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

$567+69=$

Let's solve the exercise vertically:

$567\\+69\\636$

To solve this problem, follow these steps:

• Step 1: Identify and combine the like terms in the expression $(-x) + (+2x)$.
• Step 2: Focus on the coefficients of $x$. The term $-x$ can be seen as $-1x$, and the term $+2x$ has a coefficient of $+2$.
• Step 3: Add the coefficients of these like terms: $-1 + 2 = 1$.

Now, let's simplify:

Starting with the expression:
$(-x) + (+2x) = -1x + 2x$.
Combine the coefficients: $(-1 + 2)x = 1x$.

This simplifies to $x$, as $1x$ is simply $x$.

Therefore, the solution to the problem is $x$.

To solve this problem, we'll follow these steps:

• Step 1: Add the numerators of the fractions.
• Step 2: Keep the common denominator for the result.
• Step 3: Simplify if necessary and confirm with answer choices.

Let's work through the solution step-by-step:

Step 1: Add the numerators.

The expression is $-\frac{1}{8} + -\frac{6}{8}$.

The numerators are $-1$ and $-6$. Adding these gives:

$-1 + (-6) = -1 - 6 = -7$.

Step 2: Write the result over the common denominator.

Since the denominator is $8$, the fraction becomes $-\frac{7}{8}$.

Step 3: Verify and finalize the result.

The result $-\frac{7}{8}$ matches one of the multiple-choice options.

Therefore, the solution to the problem is $-\frac{7}{8}$.

First, let's remember the rule:

$-(+x)=-x$

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

$x-4x=$

We'll add $x$ to the number line and go 4 steps to the left:

Therefore the solution is:

$x-4x=-3x$

First let's remember the rule:

$+(+x)=+x$

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

$x+3x=$

We'll can then add $x$ to the number line values and move 3 steps to the right:

Therefore, the solution is:

$x+3x=4x$

Note the following rule:

$+(-x)=-x$

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

$-x-5x=$

We'll add negative $x$ to the number line and move 5 steps to the left:

Therefore, the solution is:

$-x-5x=-6x$

Note the following rule:

$-(+x)=-x$

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

$-x-3x=$

We'll add negative $x$ to the values on the number line and move 3 steps to the left:

Therefore, the solution is:

$-x-3x=-4x$

Note the following rule:

$-(-x)=+x$

Now let's rewrite the exercise as follows:

$-x+6x=$

Then, we will write -x on the number line and move 6 steps to the right:

Therefore, the solution is:

$-x+6x=5x$

First, locate the number $\frac{1}{4}$ on the number line and move $2\frac{3}{4}$ steps to the right from it.

This means the resulting number will be positive:

Finally, solve the exercise:

$\frac{1}{4}+2\frac{3}{4}=3$

To solve this problem, we'll follow these steps:

• Step 1: Convert $3\frac{1}{3}$ to an improper fraction.
• Step 2: Find a common denominator for $-\frac{1}{5}$ and the improper fraction.
• Step 3: Add the fractions.
• Step 4: Simplify the result, converting it back to a mixed number if necessary.

Now, let's work through each step:

Step 1: Convert $3\frac{1}{3}$ to an improper fraction.
$3\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}$

Step 2: Find a common denominator for $-\frac{1}{5}$ and $\frac{10}{3}$.
The least common denominator of 5 and 3 is 15.

Step 3: Express each fraction with the common denominator:
$-\frac{1}{5} = -\frac{3}{15}$ (multiply the numerator and denominator by 3)
$\frac{10}{3} = \frac{50}{15}$ (multiply the numerator and denominator by 5)

Step 4: Add the fractions:
$-\frac{3}{15} + \frac{50}{15} = \frac{-3 + 50}{15} = \frac{47}{15}$

Step 5: Simplify $\frac{47}{15}$ back to a mixed number if needed:
Performing the division, 47 divided by 15 is 3 with a remainder of 2.
Therefore, $\frac{47}{15} = 3\frac{2}{15}$.

Therefore, the solution to the problem is $3\frac{2}{15}$.

To solve the problem of subtracting $+7\frac{1}{4}$ from $+13\frac{1}{3}$, we follow these steps:

• Convert $13\frac{1}{3}$ to an improper fraction:

$13\frac{1}{3} = \frac{39}{3} + \frac{1}{3} = \frac{39 + 1}{3} = \frac{40}{3}$

• Convert $7\frac{1}{4}$ to an improper fraction:

$7\frac{1}{4} = \frac{28}{4} + \frac{1}{4} = \frac{28 + 1}{4} = \frac{29}{4}$

• Find a common denominator for $\frac{40}{3}$ and $\frac{29}{4}$. The least common multiple of 3 and 4 is 12.
• Convert $\frac{40}{3}$ to a fraction with a denominator of 12:

$\frac{40}{3} \times \frac{4}{4} = \frac{160}{12}$

• Convert $\frac{29}{4}$ to a fraction with a denominator of 12:

$\frac{29}{4} \times \frac{3}{3} = \frac{87}{12}$

• Subtract the fractions:

$\frac{160}{12} - \frac{87}{12} = \frac{160 - 87}{12} = \frac{73}{12}$

• Convert $\frac{73}{12}$ back to a mixed fraction:

$73 \div 12 = 6$ remainder $1$, so it equals $6\frac{1}{12}$.

Therefore, the solution to the problem is $6\frac{1}{12}$.

Let's mark minus $\frac{1}{5}$ on the number line and move $3\frac{4}{5}$ steps to the left, meaning our result will be a negative number:

Let's remember the rule:

$+(-x)=-x$

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

$-\frac{1}{5}-3\frac{4}{5}=-4$