Addition and Subtraction of Directed Numbers: Identifying and defining elements

Testing through trial and error:

Let's assume both numbers are positive: 1 and 2.

1+2 = 3

Positive result.

Let's assume both numbers are negative -1 and -2

-1+(-2) = -3

Negative result.

Let's assume one number is positive and the other negative: 1 and -2.

1+(-2) = -1

Negative result.

Let's test a situation where the value of the first number is greater than the second: -1 and 2.

2+(-1) = 1

Positive result.

That is, we can see that when both numbers are positive, or in certain types of cases when one number is positive and the other negative, the sum is positive.

To solve this problem, we need to understand the properties of positive numbers and their behavior under addition.

• Step 1: Identify the given information.
  We have two numbers, $a$ and $b$, and both are positive ($a > 0$ and $b > 0$).
• Step 2: Apply the properties of positive numbers.
  According to the properties of real numbers, if both $a$ and $b$ are positive, their sum $a + b$ is also positive.
• Step 3: Conclude based on addition of positive numbers.
  Given $a > 0$ and $b > 0$, it follows directly $a + b > 0$.

Therefore, the sum of the two positive numbers $a$ and $b$ is Positive.

We will use trial and error in order to test this:

Let's assume that the value of the positive number is greater than the value of the negative number 1 and 2.

1+(-2) = -1

The result is negative.

We will try to make the value of the second number greater than the first 2 and 1.

2+(-1)= 1

The result is positive.

That is, we can see that the result depends on the values of the two numbers, so we cannot know from the beginning what the result will be.

We will illustrate with an example:

Let's assume that a is 1 and b is -2

1+ (-2) = 
1-2 = -1

Answer: Negative

Now let's assume that a is 2

and b is -1

2+(-1) = 
2-1 = 1

Even though the operation is negative, the number remains positive.

That is, if the absolute value of the positive number (a) is greater than that of the negative (b), the result will still be positive.

As we do not have data regarding this information, it is impossible to know what the sum of a+b will be.

To answer the question, we need to remember that when there is a sequence of minus and plus or plus and minus, the result will be minus.

In other words, (-)+ is actually -.

Let's demonstrate with an example:

Let's say a is -1

and b is -2

(-1)+(-2)=-3

And the result is negative!

Let's try to switch between the numbers, and say that a is -2

and b is -1

(-2)+(-1)=-3

The result is still negative!

Therefore we can conclude that adding two negative numbers results in a negative number.

We test using an example:

We define that

a = -1

b = -2

Now we replace in the exercise:

-1-(-2) = -1+2 = 1

In this case, the result is positive!

We test the opposite case, where b is greater than a

We define that

a = -2

b = -1

-2-(-1) = -2+1 = -1

In this case, the result is negative!

Therefore, the correct solution to the whole question is: "It's impossible to know".

Let's define the two numbers as 1 and 2.

Now let's place them in an exercise:

2-1=1

The result is positive!

Now let's define the numbers in reverse as 2 and 1.

Let's place an equal exercise and see:

1-2=-1

The result is negative!

We can see that the solution of the exercise depends on the absolute value of the numbers, and which one is greater than the other,

Even if both numbers are positive, the subtraction operation between them can lead to a negative result.

We test using an example:

We define that

a = -1

b = 2

Now we replace in the exercise:

-1-(2) = -1-2 = -3

In this case, the result is negative!

We test a case where the value of b is less than a

We define that

a = -2

b = 1

-2-(1) = -2-1 = -3

In this case, the result is again negative.

Since it is not possible to produce a case where a is greater than b (because a negative number is always less than a positive number),

The result will always be the same: "negative", and that's the solution!

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

• Step 1: Identify and understand the condition $a - b < 0$.
• Step 2: Evaluate each option where both numbers are negative.
• Step 3: Check to see which one yields a negative result for $a - b$.

Now, let's work through each step:

Step 1: The problem gives us $a$ and $b$ are negative numbers, and $a - b$ must also be negative, meaning $a < b$.

Step 2: Examine the given choices:

• Choice 1: $a = -5, b = 5$: Since $b$ is positive, this pair doesn't meet the condition.
• Choice 2: $a = -5, b = -6$: Calculate $a - b = -5 - (-6) = -5 + 6 = 1$, which is positive.
• Choice 3: $a = -3, b = -1$: Calculate $a - b = -3 - (-1) = -3 + 1 = -2$, which is negative.
• Choice 4: $a = -3, b = -4$: Calculate $a - b = -3 - (-4) = -3 + 4 = 1$, which is positive.

Step 3: Based on calculations above, the only pair that results in a negative $a - b$ is Choice 3: $a = -3, b = -1$. Therefore, $a = -3, b = -1$ is the correct answer.