Verified Answer :

  • The additive inverse of a number \(x\) is the value \(y\) such that \(x + y = 0\).
  • To find the additive inverse of \(-\frac{3}{2}\), compute \(-(-\frac{3}{2})\).
  • Simplifying the expression yields \(\frac{3}{2}\).
  • The final answer is \(\boxed{\frac{3}{2}}\).

Explanation

  1. Analysis of the Additive Inverse

The problem asks for the additive inverse of the real number \(-\frac{3}{2}\). By definition, the additive inverse of a real number \(x\) is the number \(y\) such that the sum of the two numbers is equal to zero: \(x + y = 0\). This relationship implies that \(y = -x\). Given \(x = -\frac{3}{2}\), our objective is to find the value of \(y = -(-\frac{3}{2})\).

  1. Calculating the Additive Inverse

To find the additive inverse \(y\), we substitute \(x = -\frac{3}{2}\) into the equation \(y = -x\). This gives:
\[y = -\left(-\frac{3}{2}\right)\]
The double negative results in a positive value. Multiplying negative one by negative three-halves yields three-halves:
\[y = \frac{3}{2}\]
We can verify this by checking the sum:
\[x + y = -\frac{3}{2} + \frac{3}{2} = 0\]
Since the sum is zero, \(\frac{3}{2}\) is indeed the additive inverse of \(-\frac{3}{2}\).

  1. Final Answer

The additive inverse of \(-\frac{3}{2}\) is found to be \(\frac{3}{2}\).
\[\boxed{\frac{3}{2}}\]

Examples

Additive inverses are fundamental in everyday accounting and physical balancing. For instance, if you have a debt of \(\$150\), which can be represented as \(-150\), the additive inverse is \(\$150\), which represents the amount needed to pay off the debt and bring your balance to zero. Similarly, if a temperature drops by \(3.5\) degrees (represented as \(-3.5\)), an increase of \(3.5\) degrees brings the temperature back to its original state. Understanding this concept helps in keeping track of net changes in any system.