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Chapter 1: Problem 19

Find each absolute value. $$ \left|1 \frac{7}{8}\right| $$

Short Answer

Expert verified
The absolute value is \(\frac{15}{8}\).

Step by step solution

01

Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. So for any number or expression inside the absolute value bars \(|...|\), just interpret it as a positive number.
02

Convert Mixed Number to Improper Fraction

To work with \(\frac{1}{8}\), convert the mixed number into an improper fraction. \(1 \frac{7}{8} = \frac{15}{8}\).
03

Apply Absolute Value

Determine the absolute value of \( \frac{15}{8} \). Since \(\frac{15}{8}\) is already positive, the absolute value is simply \( \frac{15}{8} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Calculation
Absolute value is a key mathematical concept. It represents the distance of any number from zero on the number line. This distance is always non-negative. The absolute value of any positive number or zero is the number itself. For instance, the absolute value of 5 is \(\right|5| = 5 \). For negative numbers, you disregard the sign and consider only the magnitude. So the absolute value of \(-5\) is also \(\right|-5| = 5 \). When dealing with absolute values, it is crucial to remember that the result is always a non-negative number.
Mixed Number to Improper Fraction Conversion
Mixed numbers, like \(1 \frac{7}{8}\), combine a whole number and a fraction. To convert a mixed number into an improper fraction:
  • Multiply the whole number by the denominator of the fraction
  • Add the result to the numerator of the fraction
  • Write this sum over the original denominator
For example, with \(1 \frac{7}{8}\), you first multiply 1 by 8, getting 8. Then, add 7 to get 15. Therefore, \(1 \frac{7}{8}\) as an improper fraction is \(\frac{15}{8}\). Converting to an improper fraction makes computations simpler, especially when applying operations like finding absolute values.
Non-Negative Values
In mathematics, non-negative values are either zero or any positive number. Non-negativity is crucial when working with absolute values since absolute values transform negative numbers into their positive counterparts, effectively making all results non-negative. When solving problems with absolute values, you will never encounter negative outcomes as the function strips away any negative signs, leaving only zero or positive numbers. This feature of absolute values simplifies many calculations and ensures clarity in mathematical expressions.

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