Problem 67 - Algebra A Combined Function

Question

Find each absolute value. $$ \left|-\frac{12}{13}\right| $$

Step-by-Step Solution

Answer

The absolute value of \(-\frac{12}{13}\) is \(\frac{12}{13}\).

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, the absolute value of any number is always non-negative.

Step 2: Identify the Value to Find the Absolute Value Of

In this exercise, the number is \(-\frac{12}{13}\). This is a negative fraction.

Step 3: Apply the Absolute Value Definition

Since the absolute value makes a number non-negative, we take the negative sign off \(-\frac{12}{13}\), leaving us with just \(\frac{12}{13}\).

Step 4: Confirm that Absolute Value is Non-negative

Check that the absolute value \(\frac{12}{13}\) is indeed non-negative, which fits the definition of absolute value.

Key Concepts

Understanding the Number Line

The number line is a visual representation of numbers, including both positive and negative numbers. It helps us see the position of numbers in relation to zero.
On a number line, zero is usually placed in the middle, with positive numbers extending to the right and negative numbers to the left.

• Positive numbers are located to the right of zero.
• Negative numbers are found to the left of zero.
• Zero itself is neither positive nor negative.

When we talk about numbers such as fractions, they also find their place on this line. Understanding their position helps grasp their values relative to each other. For instance, a value such as -\(\frac{12}{13}\) would be situated to the left of zero since it's a negative fraction. This visualization aids in understanding mathematical concepts like absolute value.

What is a Negative Fraction?

A negative fraction is simply a fraction where either the numerator or the denominator has a negative sign, but not both. This makes the entire fraction negative.

Consider -\(\frac{12}{13}\):

• The '-' in front indicates it's negative.
• The fraction itself means "12 out of 13," with a negative sign showing it falls below zero on the number line.

Negative fractions can be tricky in calculations, but they adhere to standard fraction rules. When dealing with the absolute value of a negative fraction, we focus on the magnitude of this fraction without considering the negative sign. This is key in transforming a negative fraction to a non-negative one when applying absolute value.

Understanding Non-negative Numbers

Non-negative numbers are numbers that are either greater than or equal to zero. These include all positive numbers and zero itself.

• Non-negative numbers never dip below zero on the number line.
• Zero is included in this category, making it distinct from only positive figures.
• Absolute values of numbers result in non-negative numbers. For instance, the absolute value of -\(\frac{12}{13}\) is \(\frac{12}{13}\).

The concept of non-negative numbers is fundamental in various fields of mathematics, aiding in defining different mathematical properties and functions. Through understanding non-negative numbers, we grasp how absolute value helps us generalize numbers irrespective of their initial negative status.

Techniques for Math Problem-Solving

Math problem-solving involves a systematic approach to tackle exercises and arrive at the correct solution. Here are some techniques:

• **Understand the problem thoroughly.** Grasp what is being asked before solving. Identify the main concepts involved.
• **Break down the problem into manageable steps.** Tackle each step individually to avoid becoming overwhelmed.
• **Double-check your work.** After solving, reassess to confirm correctness. Ensure all calculations lead to a logical conclusion.

For problems involving absolute value, remember:

• Focus on the number line. Visualize where the number lies relative to zero.
• Convert negative numbers to their positive counterparts to get non-negative results.

By following these tips, anyone can enhance their problem-solving skills and develop confidence in handling mathematical challenges.