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Chapter 1: Problem 20

Find the opposite, or additive inverse. $$ \frac{7}{2} $$

Short Answer

Expert verified
The additive inverse of \( \frac{7}{2} \) is \( -\frac{7}{2} \).

Step by step solution

01

Identify the Given Number

The given number is \( \frac{7}{2} \). To find the opposite, or additive inverse, we need to determine what number, when added to \( \frac{7}{2} \), will result in zero.
02

Understand the Concept of Additive Inverse

The additive inverse of a number is a number that, when added to the original number, equals zero. For a number \( x \), its additive inverse is \( -x \).
03

Calculate the Opposite

To find the additive inverse of \( \frac{7}{2} \), simply multiply by \( -1 \):\[ -1 \times \frac{7}{2} = -\frac{7}{2} \]
04

Verify the Solution

Add the original number and its additive inverse to verify:\[ \frac{7}{2} + \left(-\frac{7}{2}\right) = 0 \]This confirms that the additive inverse of \( \frac{7}{2} \) is \( -\frac{7}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

opposite number
Understanding the concept of an opposite number, also known as the additive inverse, can be very useful in algebra. An opposite number is simply a number that, when added to the original number, results in zero. Think about it as balancing a see-saw. If you have a weight of 5 on one side, you would need a weight of -5 on the other side to balance it out.
The opposite number of any given number x is -x. For example, if you have 3, its opposite number is -3. The same principle applies to fractions. If our number is a fraction like \(\frac{7}{2}\), its opposite is \(-\frac{7}{2}\). This inverse works perfectly to cancel out the original fraction, bringing the sum to zero.
algebraic concepts
Algebra helps us understand relationships between numbers. It's like cracking a code to find unknown values. One key concept in algebra is finding the additive inverse or opposite number.
This involves a couple of steps:
  • Identify the given number
  • Determine its opposite
For example, let鈥檚 say the given number is \(\frac{7}{2}\). To find its opposite, you multiply by -1, resulting in \(-\frac{7}{2}\). Understanding these steps makes working with equations simpler because you can easily cancel out numbers by adding their inverses.
By learning these basic algebraic rules, you build a strong foundation for more complex problems later on.
fractions
Fractions represent parts of a whole and might look tricky at first, but they follow the same rules as whole numbers, including the concept of opposite numbers.
A fraction like \(\frac{7}{2}\) can be read as 'seven halves.' Its opposite is found by multiplying by -1, to get \(-\frac{7}{2}\). This flips the fraction to the negative side of the number line.
Fractions are important in algebra because they appear in various problems and equations. Understanding how to handle them, especially when finding their additive inverse, is critical for solving many types of mathematical challenges. This makes balancing equations and simplifying expressions much easier.

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