Problem 48 - Prealgebra and Introductory Algebra

Question

What is 17 added to \(-7 ?\)

Step-by-Step Solution

Answer

The answer is \(10\)

Step 1: Understand the Problem

The problem is asking to add 17, which is a positive number, to -7, which is a negative number.

Step 2: Translate to Subtraction

This can be translated into a subtraction problem, where 17 is being subtracted by the absolute value of -7. Therefor, the problem becomes \(17 - 7\)

Step 3: Perform Subtraction

Proceed with the subtraction operation. Subtract 7 from 17 to get the result.

Key Concepts

Addition of Integers

When we talk about adding integers, we are simply combining two whole numbers that can be positive, negative, or zero. This combination can sometimes be straightforward, like adding two positive numbers or two negative numbers.
However, it can also become a bit tricky when dealing with a positive and a negative number together, as in the exercise above where you add 17 to \(-7\).

• To "add" these integers, you can rethink the problem as moving on a number line.
• Starting at the positive integer - move to the right.
• Starting at the negative integer - move to the left.

This concept is fundamental in mathematics and forms the basis of understanding more complex operations. Think of adding integers as simply counting up when numbers are positive, and counting down when they are negative. Tackling addition problems effectively often comes down to being able to confidently adjust for these leaps and bounds on your number line.

Subtraction as Inverse Operation

Subtraction and addition are inverse operations. This means they can effectively undo each other. If you add something and then subtract the same amount, you'll end up where you started.
When faced with a problem like adding a negative number, it helps to see subtraction as the addition of a negative.

• As in the above example: adding \(17\) to \(-7\) can be considered subtracting the absolute value of \(-7\) from \(17\).
• Thus, \(17 + (-7) = 17 - 7\).

Remembering that subtraction essentially reverses addition helps simplify problems by transforming them into more intuitive forms. This can make complex calculations feel less daunting and are key to mastering integer operations.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative number. Think of it like the number of steps you take away from zero, without worrying about whether you're moving into positive or negative territory.
For instance, the absolute value of both 7 and \(-7\) is 7.

• This is crucial in understanding integer addition and subtraction.
• When changing subtraction to addition as seen with \(-7\) becoming \(-) (7\), we focus solely on the magnitude.

The concept of absolute value ensures that you can work with integers effectively, helping to neutralize any confusion between positive and negative values when combining them. It's a handy tool, serving as a sort of neutralizer in calculations.

Negative Numbers

Negative numbers are numbers less than zero. They have unique properties and behave differently compared to their positive counterparts. Negative numbers represent the idea of having less than nothing or being in debt.
When operating with negatives:

• Add a negative number: it decreases the overall value.
• Subtract a negative number: it actually adds to the overall value, because subtracting a negative is akin to adding its positive counterpart.

You can visualize this conceptually on a number line, where negative moves left and positive moves right. In operations, negative numbers require careful management to ensure calculations are accurate.
Understanding these unique properties and behaviors will help in handling more complex mathematical problems, where sign errors often lead to confusion or incorrect results. Learning to work with negative numbers in operations such as the example problem is crucial for mathematical proficiency.