Problem 79 - Prealgebra

Question

Perform the indicated operations. $$12-17$$

Step-by-Step Solution

Answer

The result is -5.

Step 1: Identify the operation

The problem given is to perform the subtraction of two numbers: 12 and 17.

Step 2: Rewrite in terms of addition

Subtraction can be rewritten as adding the negative of the number. The expression \(12 - 17\) can be rewritten as \(12 + (-17)\).

Step 3: Add the numbers

When adding \(12 + (-17)\), you combine the positive number 12 and the negative number 17. Since you're adding a negative number, you're moving left on the number line, so subtract 17 from 12, giving you -5.

Key Concepts

Negative Numbers

Negative numbers are a key part of the number system and are used to represent values less than zero. They have a minus sign in front of them, for example,

• -1
• -5
• -17

Negative numbers can sometimes seem a little tricky, but once you understand that they are simply numbers below zero, they become much easier to grasp. In our exercise, we encounter the number \((-17)\). This represents a value that is 17 units below zero. When working with negative numbers, remember that the more negative a number is, the smaller its value. For instance,

• -17 is less than -5
• -5 is less than -1

Understanding this hierarchy of negativity helps immensely when performing operations like addition and subtraction.

Number Line

A number line is a visual representation that helps to understand numbers and how they relate to one another. It stretches horizontally with zero at the center, positive numbers to the right, and negative numbers to the left. Using a number line can make operations like addition and subtraction much clearer.
When you move to the right on the number line, numbers increase. Comparing this with moving to the left where numbers decrease can assist in solving problems like \(12 - 17\), which is turned into an addition problem \(12 + (-17)\). Start at 12, then move 17 steps to the left, landing on -5. The number line shows visually why "adding a negative number" is the same as "subtracting a positive number." You decrease the starting point rather than increase.

Addition of Integers

Adding integers involves combining positive and negative whole numbers. You may start by rewriting subtraction problems as addition. For example, \(12 - 17\) is the same as \(12 + (-17)\). Creating a mental shift from subtraction to addition is helpful since adding a negative number is akin to subtraction.

• If both numbers are positive, add their absolute values.
• If both numbers are negative, add their absolute values and keep the negative sign.
• If one number is negative, subtract the smaller absolute value from the larger absolute value, and the result takes the sign of the number with the larger absolute value.

For instance, \(12 + (-17)\) makes you subtract 17 from 12, giving the result -5, since 17 is larger than 12. This process aligns with the number line method we've discussed before.