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Chapter 1: Problem 17

Evaluate. $$\left|\frac{3}{2}\right|$$

Short Answer

Expert verified
Evaluating the absolute value of the given fraction \(\left|\frac{3}{2}\right|\), we directly calculate the absolute value as the fraction is already positive and in simplest form. Therefore, \(\left|\frac{3}{2}\right| = \frac{3}{2}\).

Step by step solution

01

Understand the absolute value

Absolute value of a number can be thought of as its distance from zero on the number line. It is represented by two vertical bars surrounding the number or expression. For example, the absolute value of a number x can be written as |x|.
02

Evaluate the fraction inside the absolute value

We are given the expression \(\left|\frac{3}{2}\right|\) and we need to evaluate the fraction \(\frac{3}{2}\). Since the fraction is already in simplest form, we can directly evaluate the absolute value of this fraction:
03

Evaluate the absolute value

Now, we calculate the absolute value of the fraction: \(\left|\frac{3}{2}\right| = \left|\frac {3}{2}\right| =\frac{3}{2}\) Because the fraction is already positive, its absolute value remains the same. So, the final answer is: \(\left|\frac{3}{2}\right| = \frac{3}{2}\).

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Key Concepts

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

Fraction Simplification
Fraction simplification is the process of making a fraction as simple as possible. This often involves reducing the fraction to its lowest terms. To simplify a fraction, divide both the numerator (top number) and the denominator (bottom number) by their greatest common divisor. However, in some cases, such as \(\frac{3}{2}\), the fraction may already be in its simplest form. When a fraction is in simplest form, it can鈥檛 be reduced any further while maintaining the same value. It's crucial to identify when a fraction is already simplified to avoid unnecessary calculations.
Number Line
The number line is a visual representation of numbers in a straight line, where each point corresponds to a number. It's incredibly useful for understanding concepts like absolute value. On a number line, numbers increase as you move from left to right and represent values above zero. Negative numbers are on the left of zero, while positive numbers are on the right. Absolute value is the distance from zero on a number line, so \(\left|x\right|\) shows how far \(x\) is from zero, irrespective of direction. By visualizing \(\frac{3}{2}\) on a number line, we see it is to the right of zero, confirming its positive nature.
Evaluating Expressions
Evaluating expressions involves simplifying them to a single value or more straightforward form. To evaluate an expression correctly, particularly those involving absolute values, follow these steps:
  • Identify the numbers or variables involved.
  • Simplify fractions or other mathematical operations as needed.
  • Apply mathematical rules, like the absolute value, to the simplified expression.
For \(\left|\frac{3}{2}\right|\), evaluating means recognizing that since the fraction is already positive, the absolute value leaves it unchanged. Thus, the expression evaluates directly to \(\frac{3}{2}\). It's essential to follow each step carefully to ensure accuracy.
Positive Numbers
Positive numbers are any numbers greater than zero. They are found to the right side of zero on a number line. These numbers can be whole numbers, fractions, or decimals. In mathematics, positive numbers are key in understanding other concepts, including absolute value. \(\left|x\right|\) for a positive number, \(x\), results in the same number because it is already at the needed non-negative form.
  • Examples of positive numbers include 1, 5, \(\frac{1}{2}\), and \(\frac{3}{2}\).
  • They are used in everyday calculations and measurements.
Recognizing positive numbers and their properties helps in various mathematical evaluations and operations.

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