Question

Evaluate the expression. (Review 2.1 ) $$\left|\frac{9}{10}\right|$$

Short answer

\(\frac{9}{10}\)

Step-by-step solution

Identify the absolute value expression

The problem presents the expression \( \left| \frac{9}{10} \right| \), which is an absolute value expression.

Apply the absolute value rule

The absolute value of a real number is its distance from zero on the number line. The distance is always positive or zero, never negative. Therefore, \( \left| \frac{9}{10} \right| = \frac{9}{10} \).

Key concepts

Absolute Value

The concept of absolute value is central to understanding how to evaluate expressions like the one in our exercise, which involves calculating the absolute value of a fraction, \( \left| \frac{9}{10} \right| \). In its essence, the absolute value function measures how far a number is from zero on the number line, regardless of direction. Absolute values are represented by two vertical bars surrounding the number or expression in question.

When dealing with absolute value, it is essential to remember that the result is always a non-negative number. Why is this the case? Because the absolute value reflects distance, and distance cannot be negative. Imagine you have a starting point, and you're describing how far you've moved in either direction; the distance traveled is the same whether you've moved left or right. In mathematical terms, for any real number \(x\), the absolute value is defined as follows:

\[ \left| x \right| = \begin{cases} x, & \text{if } x \geq 0 \ -x, & \text{if } x < 0 \end{cases} \]

So, for our fraction \( \frac{9}{10} \), which is a positive number, the absolute value is simply the number itself without any changes, resulting in \( \left| \frac{9}{10} \right| = \frac{9}{10} \). If you encounter a negative number inside the absolute value bars, you would negate it to find the absolute value, turning it positive.

Number Line

A number line is a visual representation of numbers laid out on a straight line, typically with zero at the center, positive numbers extending to the right, and negative numbers extending to the left. It offers a clear and simple way to understand the concept of absolute value, as it allows us to visualize the 'distance' from zero each number represents. This visualization helps reinforce why absolute value is always a positive number, as distance is direction-agnostic.

When evaluating an absolute value expression like \( \left| \frac{9}{10} \right| \), envision the fraction \( \frac{9}{10} \) positioned slightly left of 1 on the number line. This location reflects the fact that the fraction is positive and less than one whole unit from zero. To find the absolute value, simply measure how far the number is from zero. This distance is the absolute value we are looking for.

Understanding how to use a number line can significantly improve your ability to evaluate absolute value expressions quickly and accurately. It's also beneficial when learning about other mathematical concepts such as adding, subtracting, or comparing numbers.

Positive Numbers

Positive numbers are numbers greater than zero. They appear to the right of zero on the number line and are integral to many mathematical activities. In the context of absolute values, understanding positive numbers is quite straightforward: when the absolute value of a number is taken, and that number is already positive, it remains unchanged because the distance from zero is already a positive value.

So, when we look at the expression \( \left| \frac{9}{10} \right| \), we immediately recognize \( \frac{9}{10} \) as a positive number. Since absolute value expressions strip away the sign and focus on distance, the positive number within the absolute value bars indicates that there's no need for modification – it is its own absolute value. Remember that the concept of absolute value doesn't change the magnitude of a positive number; it simply assures that we are working with non-negative numbers at all times.