Problem 38
Question
For the following exercises, perform the indicated operations. $$ -15-(-10) $$
Step-by-Step Solution
Verified Answer
Answer: The result of the operation \(-15 - (-10)\) is \(-5\).
1Step 1: Understand the problem
In this exercise, we are asked to perform the subtraction: \(-15 - (-10)\). It involves subtracting a negative number from another negative number. We will use the rule that subtracting a negative number is the same as adding its opposite (positive).
2Step 2: Rewrite the expression
We rewrite the given expression, replacing the subtraction of a negative number with the addition of its opposite (positive): \(-15 - (-10)\) becomes \(-15 + 10\).
3Step 3: Perform the addition
Now, we have the expression \(-15 + 10\). Since they have different signs, we will subtract their absolute values and use the sign of the larger number in the result.
The absolute values are:
|-15| = 15
|10| = 10
Subtract the smaller absolute value from the larger: 15 - 10 = 5
The larger number in the original problem is -15, so the result will be negative. Therefore, our answer is \(-5\).
4Step 4: Final Answer
The result of the operation \(-15-(-10)\) is \(-5\).
Key Concepts
Understanding Integer OperationsDemystifying Absolute ValueAdding Opposite Integers
Understanding Integer Operations
When it comes to integer operations, we're delving into the world of both positive and negative numbers, which are the building blocks of algebra and other advanced areas of math. Integer operations include addition, subtraction, multiplication, and division with negative and positive numbers.
With subtraction of negative numbers, there's an essential rule to remember: subtracting a negative number is the same as adding its positive counterpart. So, whenever you see \( -a - (-b) \) where \( a \) and \( b \) are positive integers, you can convert it to \( -a + b \) by adding the opposite of \( b \) instead of subtracting it.
Let's consider how we can apply this to our problem \( -15 - (-10) \) from the exercise. By applying the integer operation rule, this problem simplifies to \( -15 + 10 \). The challenge then becomes adding integers with different signs, but with our rule in hand, the problem isn't as intimidating!
With subtraction of negative numbers, there's an essential rule to remember: subtracting a negative number is the same as adding its positive counterpart. So, whenever you see \( -a - (-b) \) where \( a \) and \( b \) are positive integers, you can convert it to \( -a + b \) by adding the opposite of \( b \) instead of subtracting it.
Let's consider how we can apply this to our problem \( -15 - (-10) \) from the exercise. By applying the integer operation rule, this problem simplifies to \( -15 + 10 \). The challenge then becomes adding integers with different signs, but with our rule in hand, the problem isn't as intimidating!
Demystifying Absolute Value
The absolute value of a number represents its distance from zero on a number line, without considering direction. It is always a non-negative number. Absolute value is denoted by vertical bars around the number, like so: \( |a| \).
An exemplary way to think about absolute value is to imagine it as a GPS system, which is only interested in how far you've traveled, not whether you've traveled north or south. This concept plays a crucial role when subtracting numbers with different signs. When you subtract one integer from another, you're essentially finding the distance between them.
In our step 3 from the solution, we calculated absolute values: \( |-15| = 15 \) and \( |10| = 10 \). Then, we subtracted the smaller absolute value from the larger one: \( 15 - 10 = 5 \). By understanding absolute value, we arrive at the correct magnitude of our answer, and we can use the sign of the original integer with the greater magnitude to determine the final sign.
An exemplary way to think about absolute value is to imagine it as a GPS system, which is only interested in how far you've traveled, not whether you've traveled north or south. This concept plays a crucial role when subtracting numbers with different signs. When you subtract one integer from another, you're essentially finding the distance between them.
In our step 3 from the solution, we calculated absolute values: \( |-15| = 15 \) and \( |10| = 10 \). Then, we subtracted the smaller absolute value from the larger one: \( 15 - 10 = 5 \). By understanding absolute value, we arrive at the correct magnitude of our answer, and we can use the sign of the original integer with the greater magnitude to determine the final sign.
Adding Opposite Integers
Adding opposite integers might seem puzzling at first, but it's pretty straightforward when broken down. Opposites are integers that are the same distance from zero on the number line but in opposite directions. For instance, 3 and -3 are opposites.
In terms of adding opposites, we use a key strategy: when you add a number and its opposite, the result is zero. This is because they cancel each other out. If you extend this concept, you realize that adding a negative number is the same as subtracting its positive opposite. This is why we rewrote our original problem, changing the subtraction of a negative number to the addition of a positive number.
After rewriting the problem and changing \( -15 - (-10) \) to \( -15 + 10 \), we simply followed the rules of combining integers. We added the opposite integers, and then applied the concept of absolute values to find our final answer, \( -5 \). This approach simplifies complex-looking problems and makes them more manageable.
In terms of adding opposites, we use a key strategy: when you add a number and its opposite, the result is zero. This is because they cancel each other out. If you extend this concept, you realize that adding a negative number is the same as subtracting its positive opposite. This is why we rewrote our original problem, changing the subtraction of a negative number to the addition of a positive number.
After rewriting the problem and changing \( -15 - (-10) \) to \( -15 + 10 \), we simply followed the rules of combining integers. We added the opposite integers, and then applied the concept of absolute values to find our final answer, \( -5 \). This approach simplifies complex-looking problems and makes them more manageable.
Other exercises in this chapter
Problem 38
Write the expressions for the following problems using only positive exponents. $$ \frac{1}{x^{-4}} $$
View solution Problem 38
Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ x^{3} y^{2} z^{-6} $$
View solution Problem 38
Find the sums. \(10+(-2)\)
View solution Problem 38
Determine each of the values, \(-(-|2|)\)
View solution