Problem 19 - Elementary and Intermediate Algebra

Question

Find the opposite, or additive inverse. $$ -\frac{11}{2} $$

Step-by-Step Solution

Answer

The opposite of \(-\frac{11}{2}\) is \(\frac{11}{2}\).

Step 1: Understand the Concept of Opposite or Additive Inverse

The opposite, or additive inverse, of a number is what you add to it to get zero. For any number x, the opposite (additive inverse) is -x.

Step 2: Identify the Given Number

The given number is \(-\frac{11}{2}\).

Step 3: Determine the Opposite Number

To find the opposite, change the sign of the given number. So, the opposite of \(-\frac{11}{2}\) is \(\frac{11}{2}\).

Step 4: Verify the Result

Check that the sum of the original number and its opposite is zero: \(-\frac{11}{2} + \frac{11}{2} = 0\). This confirms that \(\frac{11}{2}\) is indeed the opposite of \(-\frac{11}{2}\).

Key Concepts

opposite number

To understand the idea of an 'opposite number' or 'additive inverse,' picture a number line. Each number has a 'mirror' counterpart on the other side of zero. This opposite number, when combined with the original number, results in zero. In mathematical terms, if you have a number \( x \), its opposite is written as \( -x \). For example, the opposite of 3 is -3 because 3 + (-3) = 0. This rule applies to all types of numbers including fractions, decimals, and even negative numbers.

identifying given numbers

The first step in finding an opposite number is to identify the given number. This is straightforward but essential. In our example, the given number is \( -\frac{11}{2} \). Recognizing this number correctly is crucial because any error will lead you to an incorrect opposite number. Always double-check the original number before proceeding to find its opposite.

changing sign

Changing the sign of a number sounds like a small step but it's a powerful concept. To find the opposite, you simply swap the sign. If the number is positive, make it negative, and vice versa. For instance, to find the opposite of \( -\frac{11}{2} \), you change it to \( \frac{11}{2} \). The original number \( -\frac{11}{2} \) and its opposite \( \frac{11}{2} \) add up to zero, confirming you did it correctly. This step confirms that the opposite number is accurately found by changing the sign.