Problem 18 - Elementary and Intermediate Algebra

Question

Find the opposite, or additive inverse. $$ -17 $$

Step-by-Step Solution

Answer

The additive inverse of -17 is 17.

Step 1 - Understand Additive Inverse

The additive inverse of a number is a number that, when added to the original number, yields zero. In other words, if you have a number ‘a’, its additive inverse is ‘-a’ such that: a + (-a) = 0

Step 2 - Determine the Additive Inverse

For the given number -17 we need to find its additive inverse. Adding 17 to -17 should result in zero. Thus, the additive inverse of -17 is 17.

Step 3 - Verification

Verify the result by adding the original number to its additive inverse: -17 + 17 = 0. Since the sum is zero, 17 is indeed the additive inverse of -17.

Key Concepts

opposite number

When we talk about the 'opposite number' in mathematics, we often refer to the concept of the additive inverse. The opposite number of a particular number is the one that, when added to the original number, results in zero. Imagine standing at a point on a number line. If you want to return to the center (which is zero), you have to move an equal distance in the opposite direction.
For instance, consider the number 5. The opposite number is -5, because:

• 5 + (-5) = 0

The same logic applies to negative numbers. If we start with -17, its opposite number is 17. Together they bring us back to zero:

• -17 + 17 = 0

This concept is fundamental, especially in basic algebra and when solving equations.

negative numbers

Negative numbers are numbers that are less than zero. They usually have a minus sign (-) in front of them. Negative numbers are often seen in various contexts, such as temperatures below freezing, or financial debts. Visualize a number line with zero in the center:

• Positive numbers are to the right of zero
• Negative numbers are to the left of zero

Negative numbers have interesting properties. For instance:

• Adding a negative number is the same as subtracting its positive counterpart, e.g., 7 + (-4) = 7 - 4
• Multiplying two negative numbers yields a positive result, e.g., (-3) * (-2) = 6
• Adding two negative numbers results in a more negative number, e.g., (-3) + (-2) = -5

Understanding negative numbers is crucial for mastering basic algebra.

basic algebra

Basic algebra involves understanding how to manipulate numbers and variables to solve equations. One core concept in algebra is using additive inverses to isolate variables and solve equations. Let's consider a simple equation:

• x + 5 = 12

To isolate x, we use the opposite number of 5, which is -5:

• x + 5 - 5 = 12 - 5
• x = 7

In this equation, we subtracted 5 from both sides to isolate x.
Another crucial aspect of algebra is understanding operations involving negative numbers. For instance, solving a - x = b requires knowing that moving x to the other side of the equation changes its sign. For example:

• 7 - x = 3
• -x = 3 - 7
• -x = -4
• x = 4 (multiplying both sides by -1)

These foundational skills are necessary for more advanced algebraic equations and concepts.