Problem 34 - Algebra 2

Question

Find each absolute value. $$ \left|-\frac{2}{3}\right| $$

Step-by-Step Solution

Answer

\( \left| -\frac{2}{3} \right| = \frac{2}{3} \).

Step 1: Understand Absolute Value

The absolute value of a number is the distance it is from zero on the number line, which means it is always non-negative. No matter if the value inside the absolute operator is positive or negative, the output will always be positive.

Step 2: Simplify the Expression

We have the absolute value expression \( \left| -\frac{2}{3} \right| \). Since the absolute value makes any negative value positive, we remove the negative sign. Thus, \( \left| -\frac{2}{3} \right| = \frac{2}{3} \).

Key Concepts

Number Line

Imagine a long horizontal line filled with numbers. This is the 'number line'. It's like a ruler, stretching infinitely in both directions, with zero right smack in the middle. Positive numbers are on the right, while negative numbers are on the left. Each point on this line represents a number. Understanding this layout is a key part of grasping absolute value. Once a number is plotted on this line, its distance from zero tells us its absolute value. The further away from zero, the larger the absolute value is, regardless of direction.

Working with a number line makes it easy to visualize distance, helping us see why the absolute value is never negative. Even if a number is negative, like \( -\frac{2}{3} \), the absolute value, or the distance from zero, remains positive at \( \frac{2}{3} \).

• Zero marks the center of the number line.
• Numbers to the right are positive.
• Numbers to the left are negative.

Non-negative Numbers

Non-negative numbers are simply numbers that are either positive or zero. They never drop below zero. This concept is vital in understanding absolute values. When you're dealing with absolute values, like \(\left| -\frac{2}{3} \right|\), you know the outcome will always be a non-negative number. That's because absolute value basically measures the how-far-from-zero factor on the number line. A quick checklist for non-negative numbers:

• Zero is non-negative.
• Any positive number is non-negative.
• No negatives allowed!

Non-negative numbers help ensure our results in many mathematical contexts stay within the realm of the possible.

Simplifying Expressions

Simplifying expressions involves making complex mathematical phrases easier to understand and solve. This process is like tidying up a messy room - you're making it clear and simple to use. When we simplify the expression \(\left| -\frac{2}{3} \right|\), we're determining its absolute value. Since absolute value makes any negative component into its positive counterpart, the simplified expression becomes \(\frac{2}{3}\). Key steps for simplifying expressions:

• Identify operations like absolute value, which need addressing.
• Focus on removing complexity (e.g., changing negatives to positives where needed).
• Re-check if simpler, equivalent form is achieved.

Simplifying makes future operations easier and ensures the answer is in the most understandable form.