Verified Answer :

  • Determine the absolute values of the individual numbers: \(\vert-7\vert = 7\) and \(\vert-5\vert = 5\).
  • Subtract the second value from the first: \(7 - 5 = 2\).
  • The final answer is \(\boxed{2}\).

Explanation

  1. Understanding the Absolute Value Function

To evaluate the expression \(\vert-7\vert - \vert-5\vert\), we first recall the definition of the absolute value function. The absolute value of a real number \(x\), denoted by \(\vert x\vert\), represents the non-negative distance of the number from 0 on the number line. Specifically, \(\vert x\vert = x\) if \(x \geq 0\), and \(\vert x\vert = -x\) if \(x < 0\).

  1. Evaluating the Individual Terms

We evaluate the individual terms:

  • For \(\vert-7\vert\), since \(-7 < 0\), we have \(\vert-7\vert = -(-7) = 7\).
  • For \(\vert-5\vert\), since \(-5 < 0\), we have \(\vert-5\vert = -(-5) = 5\).
  1. Final Calculation

Now, we substitute these values back into the original expression:
\(\vert-7\vert - \vert-5\vert = 7 - 5 = 2\).
Thus, the final result is 2.

Examples

The concept of absolute value is essential in measuring distances or differences where the direction doesn't matter, only the magnitude. For example, if you are looking at a thermometer and the temperature drops from -5 degrees to -7 degrees, the difference in temperature magnitude is \(\vert-7 - (-5)\vert = \vert-2\vert = 2\) degrees. Similarly, if you are tracking changes in your account balance that result in debt, absolute values help you quantify the size of the changes regardless of whether the balance is positive or negative.

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