Verified Answer :

  • Evaluate the expressions to get the values: \(-7, 7, 3, -6, 0, -5\).
  • Compare the numerical values: \(-7 < -6 < -5 < 0 < 3 < 7\).
  • Map these values back to their original expressions.
  • The order from least to greatest is: \(\boxed{{-7, -|6|, -|-5|, 0, 3, |-7|}}\).

Explanation

  1. Analysis of the Expressions

We are tasked with ordering a set of six real numbers from least to greatest: \(-7\), \(|-7|\), \(3\), \(-|6|\), \(0\), and \(-|-5|\). To compare these values, we must first evaluate each expression, recalling that the absolute value function \(|x|\) returns the non-negative distance of \(x\) from zero on the number line. Specifically, \(|x| = x\) if \(x \ge 0\) and \(|x| = -x\) if \(x < 0\).

  1. Evaluating Each Term

Now we calculate the value of each term individually:

  1. \(-7\) is already given as \(-7\).
  2. For \(|-7|\), the absolute value of \(-7\) is \(7\).
  3. For \(3\), the value is simply \(3\).
  4. For \(-|6|\), we first find \(|6| = 6\), then negate it to obtain \(-6\).
  5. For \(0\), the value is \(0\).
  6. For \(-|-5|\), we first find \(|-5| = 5\), then negate it to obtain \(-5\).

The set of evaluated numbers is \(\{-7, 7, 3, -6, 0, -5\}\).

  1. Ordering the Values

To order the set \(\{-7, 7, 3, -6, 0, -5\}\) from least to greatest, we compare them:

  • The smallest value is \(-7\).
  • Next is \(-6\), which corresponds to \(-|6|\).
  • Next is \(-5\), which corresponds to \(-|-5|\).
  • Next is \(0\).
  • Next is \(3\).
  • The largest value is \(7\), which corresponds to \(|-7|\).

Thus, the ordered sequence is \(-7, -6, -5, 0, 3, 7\). Expressed in the original notation, this is \(-7, -|6|, -|-5|, 0, 3, |-7|\).

Examples

Comparing absolute values is essential for understanding concepts like displacement and magnitude. For example, if you track the temperature changes in a city, you might have values like \(-5^\circ C\), \(|-2^\circ C|\), and \(0^\circ C\). Converting these to their absolute magnitudes or simply comparing their positions on a number line helps in identifying the coldest or warmest days in a week. This type of comparison is a fundamental skill in data analysis and basic science.

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