Problem 58 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{4 x}{3}+\frac{5}{9}-\frac{11 x}{6}$$

Step-by-Step Solution

Answer

\( \frac{-9x + 10}{18} \) is the simplified expression.

Step 1: Identify the Operation

First, look at the expression: \(\frac{4x}{3} + \frac{5}{9} - \frac{11x}{6}\). You need to perform addition and subtraction with fractions, so our first step is to make sure to align the fractions for these operations.

Step 2: Find a Common Denominator

The denominators are 3, 9, and 6. The least common denominator (LCD) for these numbers is 18. We will rewrite each fraction with this common denominator.

Step 3: Rewrite Fractions with Common Denominator

- Convert \( \frac{4x}{3} \) to the common denominator of 18. Multiply both numerator and denominator by 6 to get: \( \frac{24x}{18} \).- Convert \( \frac{5}{9} \) to the common denominator of 18. Multiply both numerator and denominator by 2 to get: \( \frac{10}{18} \).- Convert \( \frac{11x}{6} \) to the common denominator of 18. Multiply both numerator and denominator by 3 to get: \( \frac{33x}{18} \).

Step 4: Perform the Operations

Now that all fractions have a common denominator, perform the operations: \( \frac{24x}{18} + \frac{10}{18} - \frac{33x}{18} \). Combine like terms:- Combine \( \frac{24x}{18} - \frac{33x}{18} \) to get \( -\frac{9x}{18} \).- Add \( -\frac{9x}{18} + \frac{10}{18} \) to get \( \frac{-9x + 10}{18} \).

Step 5: Simplify the Expression

The result is \( \frac{-9x + 10}{18} \). Check if the expression can be simplified further. Since there are no common factors, this is the simplest form.

Key Concepts

Common Denominator

When working with fractions, especially involving addition and subtraction, finding a common denominator is key. This step ensures that denominators match before proceeding with any calculations, allowing us to focus on the numerators.

• The common denominator is a multiple that each of the original denominators can divide into without leaving a remainder.
• To find the least common denominator (LCD), compute the least common multiple of all the denominators involved in the fractions.
• In our problem, we had denominators 3, 9, and 6. The smallest number that all these can divide evenly is 18, making it the common denominator for this equation.

By using a common denominator, the operation of adding or subtracting becomes straightforward since the focus shifts towards managing just the numerators.

Addition and Subtraction of Fractions

Once the fractions have a common denominator, we can smoothly proceed to add or subtract them. This involves only the numerators since the denominators are already aligned.

• In the worked example, after aligning with the common denominator, the expression becomes: \( \frac{24x}{18} + \frac{10}{18} - \frac{33x}{18} \).
• Add or subtract the numerators: \( 24x - 33x \) for the terms with \( x \), and then add \( 10 \) for the constant.
• This results in combining like terms, resulting in \( -9x + 10 \) over the common denominator of 18.

Remember, when adding fractions, numerators are combined according to their respective signs, ensuring mathematical accuracy.

Simplifying Algebraic Expressions

The final step in working with algebraic fractions is simplifying the result. Simplification involves reducing fractions to their most basic form, ensuring no further numerical factorization is possible.

• Check if the numerator and denominator share any common factors which can be divided out.
• In the given exercise, the expression obtained is \( \frac{-9x + 10}{18} \). Here, no common factor exists between \(-9x + 10\) and 18, as the terms do not share variables or numbers that factor evenly.
• Thus, \( \frac{-9x + 10}{18} \) is already in its simplest form. If there were common factors, they would be divided out to reduce further complexity.

Simplification is crucial for making expressions less cumbersome and easier to interpret or use in further calculations.