Problem 66 - Algebra and Trigonometry

Question

\(61-66=\) Evaluate each expression. (a) \(\left|\frac{-6}{24}\right| \quad\) (b) \(\left|\frac{7-12}{12-7}\right|\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{4}\), (b) 1

Step 1: Evaluate the Expression in Part (a)

For part (a), we need to evaluate \( \left| \frac{-6}{24} \right| \). Start by simplifying the fraction within the absolute value.\[ \frac{-6}{24} = \frac{-1}{4} \]The absolute value of \(-\frac{1}{4}\) is \(\frac{1}{4}\) as absolute value makes the number positive.

Step 2: Evaluate the Expression in Part (b)

For part (b), evaluate the expression inside the absolute value, \( \left| \frac{7-12}{12-7} \right| \). Begin by calculating the numerator and the denominator separately: \[ 7 - 12 = -5, \quad 12 - 7 = 5 \]This gives us \( \frac{-5}{5} = -1 \). The absolute value of \(-1\) is \(1\).

Key Concepts

Fractions

Fractions are a way to represent numbers that are not whole. Essentially, they show how many parts of a whole you have. A fraction consists of two numbers separated by a line. The number above the line is called the **numerator**, and the number below the line is called the **denominator**. For example, in the fraction \( \frac{-6}{24} \), -6 is the numerator, and 24 is the denominator.
This way of representing numbers is particularly useful when one number divides another without resulting in whole numbers. Fractions can describe values between whole numbers, allowing us to express parts of a whole and perform calculations with non-integral parts. When using fractions:

• The **numerator** indicates how many parts you have.
• The **denominator** tells you how many parts make up a whole.

In everyday tasks, fractions enable us to perform operations on parts of items, such as dividing a pizza, measuring ingredients in a recipe, or dividing a sum of money among several people.

Numerator and Denominator

Understanding the **numerator** and **denominator** is crucial when working with fractions. The numerator is the top number in a fraction and tells you how many parts you are considering. The denominator is the bottom number and indicates the total number of equal parts in the whole that those numerator parts are taken from.For example, in part (b) of the exercise, we have the fraction \( \frac{7-12}{12-7} \). To solve it:

• Calculate the numerator: \( 7 - 12 = -5 \).
• Calculate the denominator: \( 12 - 7 = 5 \).

Thus, we end up with the fraction \( \frac{-5}{5} \).Knowing how to find and use these two numbers correctly is essential when simplifying, comparing, or evaluating fractions. Fractions can represent a whole when the numerator is equal to the denominator, as in this example where \( \frac{-5}{5} = -1 \). This particular form of a fraction indicates that the parts in the numerator totally make up the whole, simplified down to -1.

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics that involves reducing fractions or algebraic statements to their simplest form. This process requires recognizing and eliminating any unnecessary complexity in an expression.In the original exercise, simplifying the fraction and taking the absolute value was essential. For part (a), the expression \( \left| \frac{-6}{24} \right| \) simplifies as follows:

• Simplify the fraction: \( \frac{-6}{24} \) divides both the numerator and denominator by 6, resulting in \( \frac{-1}{4} \).
• Take the absolute value: The absolute value changes \(-\frac{1}{4}\) to \(\frac{1}{4}\).

Part (b) involves more steps but follows the same principle of reducing complexity:

• Simplify the fraction: Calculate \( \frac{7-12}{12-7} \) to get \( \frac{-5}{5} \), which simplifies to \(-1\).
• Take the absolute value: Absolute value changes \(-1\) to 1.

Essentially, simplifying means making calculations straightforward and easy to interpret. It results in cleaner and more manageable expressions, essential for solving complex problems accurately.