Problem 58 - Algebra and Trigonometry

Question

Evaluate each expression. $$ \begin{array}{ll}{\text { (a) }\left|\frac{-6}{24}\right|} & {\text { (b) }\left|\frac{7-12}{12-7}\right|}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Evaluate Expression (a)

For part (a), we need to evaluate the absolute value of \( \frac{-6}{24} \). First, simplify \( \frac{-6}{24} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 6. This gives us \( \frac{-1}{4} \). Thus, the expression becomes \( |\frac{-1}{4}| \).

Step 2: Calculate Absolute Value of (a)

The absolute value function removes any negative sign, turning negative numbers into positive ones. Hence, \( |\frac{-1}{4}| = \frac{1}{4} \).

Step 3: Evaluate Numerator for Expression (b)

For part (b), evaluate the numerator \( 7 - 12 \), which equals \(-5\).

Step 4: Evaluate Denominator for Expression (b)

Evaluate the denominator \( 12 - 7 \), which equals \( 5 \).

Step 5: Simplify Fraction in (b)

Now, substitute the results into the fraction: \( \frac{-5}{5} \). Simplifying this fraction yields \(-1\).

Step 6: Calculate Absolute Value of (b)

Calculate the absolute value: \( |-1| \). The absolute value of \(-1\) is 1.

Key Concepts

Absolute Value

When you hear the term "absolute value," think of distance. Absolute value refers to how far a number is from zero on the number line, without considering direction. In simpler terms, it's about the magnitude of a number. Whether the number is positive or negative, the absolute value is always non-negative.

For example:

• The absolute value of \(-3\) is 3 because it is three steps away from zero.
• The absolute value of 4 is also 4 because it doesn't change with direction.

In the given exercise, the absolute value helps simplify \( \left| \frac{-1}{4} \right| \) and \( \left| -1 \right| \). Both expressions become positive as absolute value strips away any negative signs.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. This means finding an equivalent fraction where the numerator and denominator have no common factors other than 1. Fraction simplification helps in easier computation and comparison between fractions.

To simplify a fraction, you:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both numerator and denominator by the GCD.

In the exercise, \( \frac{-6}{24} \) is simplified by recognizing that 6 is the GCD, resulting in \( \frac{-1}{4} \). It then becomes straightforward to compute its absolute value.

Numerator and Denominator

Every fraction is composed of two key components:

• The numerator, which is the top number and indicates how many parts you have.
• The denominator, which is the bottom number and shows the total number of equal parts into which the whole is divided.

Understanding these parts is crucial when evaluating fractions. For example, in exercise (b), the expression \( \frac{7-12}{12-7} \) requires calculating both the numerator (7 - 12 = -5) and the denominator (12 - 7 = 5). The fraction becomes \( \frac{-5}{5} \). Knowing how to find each part simplifies solving the problem accurately.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. Finding the GCD is essential for shortening a fraction to its simplest form.

Here's how to find the GCD:

• List the factors of each number (those numbers that divide it evenly).
• Identify the largest common factor shared between the two lists.

In the example given, the GCD of 6 and 24 is 6. Dividing both by their GCD results in the simplified fraction \( \frac{-1}{4} \). Understanding this concept will help simplify fractions quickly and efficiently.