Problem 27 - Beginning and Intermediate Algebra

Question

Subtract. See Examples 1 through 5 $$ -\frac{3}{11}-\left(-\frac{5}{11}\right) $$

Step-by-Step Solution

Answer

The result is \(\frac{2}{11}\).

Step 1: Identify the Operation

The operation we need to perform is subtraction of two negative fractions: -\(-\frac{3}{11}\)and \(-\left(-\frac{5}{11}\right)\). When subtracting a negative number, it is equivalent to adding the positive of that number.

Step 2: Convert Subtraction to Addition

The expression \(-\frac{3}{11} - \left(-\frac{5}{11}\right)\)is equivalent to rewriting subtraction of a negative as addition, which gives \(-\frac{3}{11} + \frac{5}{11}\).

Step 3: Add the Fractions

Now, add the two fractions. Since they have the same denominator, you can simply add the numerators. \(-3 + 5 = 2\). So, \(-\frac{3}{11} + \frac{5}{11} = \frac{2}{11}\).

Step 4: Simplify the Result

Check if the fraction can be simplified. In this case, \(\frac{2}{11}\)is already in its simplest form, so no further simplification is needed.

Key Concepts

Understanding Negative Fractions

Negative fractions can be a little tricky, but they are simply fractions with a negative sign. This means the fraction is less than zero. For example,

• The fraction \(-\frac{3}{11}\) is negative because there is a minus sign in front of it.
• It represents 3 parts of 11 below zero.

When working with negative fractions, it's important to pay attention to the minus sign.
If you see an expression like \(-\left(-\frac{5}{11}\right)\), you have a double negative. In math, two negatives make a positive, just like in life! If we subtract a negative fraction, we actually add it: \(-\left(-\frac{5}{11}\right) = \frac{5}{11}\). Understanding this concept makes it easier to handle operations with negative fractions.

Adding Fractions: The Basics

Adding fractions is all about the denominators. For fractions like \(-\frac{3}{11} + \frac{5}{11}\), the denominators are the same, which makes addition straightforward. This means we can directly add their numerators:

• \(-3 + 5 = 2\)

Keep the common denominator as it is, which gives us the resulting fraction \(\frac{2}{11}\). When denominators are different, it's necessary to find a common denominator before performing the addition. You can do this by finding the least common multiple (LCM) of the denominators, converting each fraction to an equivalent fraction with this common denominator, and then adding the numerators together.

Simplifying Fractions to Their Simplest Form

Simplifying fractions means finding an equivalent fraction where the numerator and the denominator have no common factors other than 1. A fraction is in its simplest form when no further reduction is possible.

• When you have a fraction like \(\frac{2}{11}\), check if 2 and 11 have any common factors.
• Since both numbers are prime relative to each other (they do not share any other factors than 1), \(\frac{2}{11}\) is already in its simplest form.

To simplify other fractions, divide both the numerator and the denominator by their greatest common divisor (GCD). This process reduces the fraction to its lowest terms. Understanding how to simplify fractions is an essential skill that can help make working with fractions easier and more intuitive.