Problem 64 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{4}{9 x}-\frac{7}{6 x^{2}}$$

Step-by-Step Solution

Answer

\(\frac{8x - 21}{18x^2}\)

Step 1: Determine the Least Common Denominator (LCD)

To add or subtract fractions, they must have the same denominator. The denominators here are \(9x\) and \(6x^2\). The LCD is the least common multiple of these, which is \(18x^2\).

Step 2: Adjust the Fractions to Have the LCD

For the first fraction \(\frac{4}{9x}\), multiply the numerator and the denominator by \(2x\) to obtain \(\frac{4 \times 2x}{9x \times 2x} = \frac{8x}{18x^2}\). For the second fraction \(\frac{7}{6x^2}\), multiply the numerator and denominator by 3 to obtain \(\frac{7 \times 3}{6x^2 \times 3} = \frac{21}{18x^2}\).

Step 3: Perform the Subtraction

Now that the fractions have the same denominator, subtract the numerators: \(\frac{8x}{18x^2} - \frac{21}{18x^2} = \frac{8x - 21}{18x^2}\).

Step 4: Simplify the Fraction

Check if the expression \(\frac{8x - 21}{18x^2}\) can be simplified. Since there is no common factor between the numerator \(8x - 21\) and the denominator \(18x^2\), the fraction is already in its simplest form.

Key Concepts

Understanding the Least Common Denominator

When working with fractions, especially when you need to add or subtract them, it's essential to have the same denominator. This is where the concept of the Least Common Denominator (LCD) comes in. The LCD is the smallest expression that is a multiple of each of the denominators you're dealing with. For example, in our exercise with the fractions \(\frac{4}{9x}\) and \(\frac{7}{6x^2}\), the denominators are \(9x\) and \(6x^2\). To find the LCD, you identify the highest powers of all variables and any numeric factors potentially needed to cover all parties. In this case, multiplying the highest power of \(x\) which is \(x^2\) by the smallest number that \(9\) and \(6\) both divide into, we get the LCD as \(18x^2\).

• Always start by finding the greatest power of common variables.
• Then find the smallest number that all numerical factors divide into.

This process ensures that both fractions can be rewritten with this shared denominator, making addition or subtraction straightforward.

Mastering Simplifying Fractions

Simplifying fractions is about reducing them to their most basic form, where the numerator and the denominator have no common factors other than 1. In our worked example, we ended up with \(\frac{8x - 21}{18x^2}\). To simplify, you should initially factor both the numerator and the denominator. However, for the expression \(8x - 21\) and \(18x^2\), any common factors would need to be present in both.

• Check each term in the numerator for any factors it shares with the denominator.
• Factor by grouping if applicable, though in many cases like this, the simplest form is as is unless there are obvious numerical commonalities.

If no common factors exist, then the fraction already is as simplified as possible. Stay alert for common traps such as assuming terms sharing letters are simplifiable without true numerical factors.

Diving into Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols, and they form the basis of many algebraic rules like our given exercise. When fractions include algebraic expressions, you handle them using the same principles as numerical fractions but with extra attention to variables.

• Variables act as placeholders for numbers and follow the same arithmetic rules. Keeping track of these rules helps when forming new expressions needing simplification or manipulation.
• Pay close attention to both coefficients (the numerical part) and the power of any variables. For instance, in \(\frac{4}{9x}\), \(x\) acts as a core part of denominator logic.

By understanding how to work with expressions, you'll find it easier to apply these concepts flexibly across different problems. Practice is key, as getting comfortable with algebraic fractions opens up a world of more complex mathematics to explore without getting overwhelmed by the details.