- Evaluate each absolute value expression by converting the mixed numbers into improper fractions: \(A = \frac{31}{7}, B = \frac{34}{7}, C = \frac{38}{7}, D = \frac{30}{7}, E = \frac{36}{7}\).
- Compare the resulting fractions: \(\frac{30}{7} < \frac{31}{7} < \frac{34}{7} < \frac{36}{7} < \frac{38}{7}\).
- Order the expressions accordingly: \(\boxed{\left|4 \frac{2}{7}\right|, \left|-4 \frac{3}{7}\right|, \left|-4 \frac{6}{7}\right|, \left|5 \frac{1}{7}\right|, \left|-5 \frac{3}{7}\right|}\).
Explanation
- Understanding the Absolute Value and Given Expressions
The objective is to order five expressions involving absolute values of mixed numbers from least to greatest. The expression for absolute value, denoted by \(\left| x \right|\), represents the distance of a number \(x\) from \(0\) on the number line. If \(x \ge 0\), then \(\left| x \right| = x\), and if \(x < 0\), then \(\left| x \right| = -x\). This means that for any real number, the absolute value is always non-negative. We are given the following expressions:
\[A = \left|-4 \frac{3}{7}\right|, B = \left|-4 \frac{6}{7}\right|, C = \left|-5 \frac{3}{7}\right|, D = \left|4 \frac{2}{7}\right|, E = \left|5 \frac{1}{7}\right|.\]
- Evaluating the Expressions
We now calculate the numerical value of each expression. To do this, we treat each mixed number as a sum and apply the absolute value property:
- For \(A = \left|-4 \frac{3}{7}\right|\): \(\left|-\frac{28}{7} - \frac{3}{7}\right| = \left|-\frac{31}{7}\right| = \frac{31}{7}\).
- For \(B = \left|-4 \frac{6}{7}\right|\): \(\left|-\frac{28}{7} - \frac{6}{7}\right| = \left|-\frac{34}{7}\right| = \frac{34}{7}\).
- For \(C = \left|-5 \frac{3}{7}\right|\): \(\left|-\frac{35}{7} - \frac{3}{7}\right| = \left|-\frac{38}{7}\right| = \frac{38}{7}\).
- For \(D = \left|4 \frac{2}{7}\right|\): \(\left|\frac{28}{7} + \frac{2}{7}\right| = \left|\frac{30}{7}\right| = \frac{30}{7}\).
- For \(E = \left|5 \frac{1}{7}\right|\): \(\left|\frac{35}{7} + \frac{1}{7}\right| = \left|\frac{36}{7}\right| = \frac{36}{7}\).
- Comparing and Ordering the Results
We now compare the evaluated values, all of which are fractions with the same denominator \(7\). By comparing the numerators, we can easily order the fractions from least to greatest:
\[\frac{30}{7} < \frac{31}{7} < \frac{34}{7} < \frac{36}{7} < \frac{38}{7}.\]
Mapping these values back to the original expressions, we obtain the ordered list:
- \(\left|4 \frac{2}{7}\right| = \frac{30}{7}\)
- \(\left|-4 \frac{3}{7}\right| = \frac{31}{7}\)
- \(\left|-4 \frac{6}{7}\right| = \frac{34}{7}\)
- \(\left|5 \frac{1}{7}\right| = \frac{36}{7}\)
- \(\left|-5 \frac{3}{7}\right| = \frac{38}{7}\)
Examples
Understanding absolute values is essential when measuring physical quantities that cannot be negative, such as distance, speed, or temperature differences. For example, if a car travels from point A to point B and back, the displacement might be zero, but the distance traveled—which is the absolute value of the displacement at each point in time relative to the start—is always a positive, cumulative value. Mastering these comparisons helps in everyday tasks like comparing price differences, bank account balances relative to zero, or even adjusting physical measurements in construction and engineering.