Problem 20 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(17+(-17)\)

Step-by-Step Solution

Answer

0

Step 1: Understand the Problem

Identify the numbers involved in the addition. Here, the numbers are 17 and -17.

Step 2: Recognize the Opposite Numbers

Notice that 17 and -17 are opposites of each other. One is positive and the other is negative.

Step 3: Apply the Concept of Opposites

When you add a number and its opposite, the sum is 0. This is because they cancel each other out. So, adding 17 and -17 results in 0.

Step 4: Verify the Solution (Optional)

You can check your work using a number line. Start at 17 and move left 17 units, which brings you to 0.

Key Concepts

Opposite Numbers

Opposite numbers are pairs of numbers that are the same distance from zero on the number line but in opposite directions. For example, 17 and -17 are opposites. This means:

• 17 is 17 units to the right of zero
• -17 is 17 units to the left of zero

When you add opposite numbers, they cancel each other out because one is positive and the other is negative. This concept is very useful when adding integers, especially when working with both positive and negative numbers. If you understand opposite numbers, it becomes easier to see why the sum of 17 and -17 is 0.

Number Line

A number line is a visual representation of numbers in a straight line. It helps us understand and perform operations like addition and subtraction more intuitively. Imagine the number line with zero in the middle. Positive numbers are placed to the right, and negative numbers are to the left. Here’s how the number line helps in our problem:

• Starting at 0, moving 17 units to the right places you at 17.
• Moving 17 units to the left from 17 brings you back to 0.

This shows that 17 and -17 will indeed sum to zero. Using the number line is a great way to check your work and ensure your calculations are correct. It provides a visual method to grasp how addition of positive and negative integers works.

Integer Addition

Integer addition involves combining positive and negative whole numbers. Here are some important rules to remember:

• Adding two positive integers results in a larger positive integer. For example, 5 + 3 = 8.
• Adding two negative integers results in a larger negative integer in terms of absolute value. For example, -5 + (-3) = -8.
• Adding a positive integer and a negative integer involves finding the difference between their absolute values and then assigning the sign of the number with the larger absolute value. For example, 6 + (-4) = 2, and -6 + 4 = -2.

Understanding these rules helps simplify complex problems. Always start by identifying the signs and magnitudes of the numbers involved. This clarity ensures you follow the right steps in adding integers, whether positive or negative.