Verified Answer :

  • Identify the absolute value of each number: \(|-4| = 4\) and \(|6| = 6\).
  • Add the resulting positive values together: \(4 + 6 = 10\).
  • The simplified result is \(\boxed{10}\).

Explanation

  1. Analyzing the Absolute Value Expression

The goal is to simplify the expression \(|-4| + |6|\). The absolute value of a real number, denoted by \(|x|\), is the distance of the number from zero on the number line. Specifically, it is defined as:

  • \(|x| = x\) if \(x \ge 0\)
  • \(|x| = -x\) if \(x < 0\)

This means the absolute value is always non-negative. We will apply this definition to each term separately.

  1. Calculating the Sum

First, consider the term \(|-4|\). Since \(-4 < 0\), we apply the second part of the definition, which states \(|-4| = -(-4) = 4\).

Next, consider the term \(|6|\). Since \(6 > 0\), we apply the first part of the definition, which states \(|6| = 6\).

Now, we add these two results together to find the value of the entire expression:
\[|-4| + |6| = 4 + 6\]

  1. Final Conclusion

Finally, we perform the addition: \(4 + 6 = 10\). Therefore, the expression simplifies to 10.
\[|-4| + |6| = \boxed{10}\]

Examples

The concept of absolute value is essential in measuring distances or differences where the sign does not matter. For example, if you are calculating the difference in temperature between a cold morning at \(-4^\circ\text{C}\) and a warm afternoon at \(6^\circ\text{C}\), you calculate the total change in temperature as the absolute distance between these points on a thermometer: \(|-4 - 6| = |-10| = 10^\circ\text{C}\). Absolute values ensure that we always talk about positive magnitudes when measuring such deviations or distances.

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