Question

Simplify (a) \(\frac{2}{x}-\frac{3}{2 x}\), (b) \(\frac{2}{x}+\frac{3}{2 x}\), (c) \(\frac{2}{x} \times \frac{3}{2 x}\).

Short answer

Question: Simplify the following expressions: (a) \(\frac{2}{x}-\frac{3}{2 x}\), (b) \(\frac{2}{x}+\frac{3}{2 x}\), (c) \(\frac{2}{x} \times \frac{3}{2 x}\). Answer: (a) \(\frac{1}{2x}\), (b) \(\frac{7}{2x}\), (c) \(\frac{3}{x^2}\)

Step-by-step solution

Simplifying Expression (a)

To simplify the expression \(\frac{2}{x}-\frac{3}{2 x}\), first find the common denominator, which is \(2x\). Then, rewrite the expression with common denominators and subtract: \(\frac{2}{x}-\frac{3}{2 x} = \frac{4}{2x}-\frac{3}{2 x}\). Now, subtract the numerators while keeping the same denominator: \(\frac{4}{2x}-\frac{3}{2 x} = \frac{4-3}{2x}\). Finally, simplify: \(\frac{4-3}{2x} = \frac{1}{2x}\).

Simplifying Expression (b)

To simplify the expression \(\frac{2}{x}+\frac{3}{2 x}\), first find the common denominator, which is \(2x\). Then, rewrite the expression with common denominators and add: \(\frac{2}{x}+\frac{3}{2 x} = \frac{4}{2x}+\frac{3}{2 x}\). Now, add the numerators while keeping the same denominator: \(\frac{4}{2x}+\frac{3}{2 x} = \frac{4+3}{2x}\). Finally, simplify: \(\frac{4+3}{2x} = \frac{7}{2x}\).

Simplifying Expression (c)

To simplify the expression \(\frac{2}{x} \times \frac{3}{2 x}\), multiply the numerators together and the denominators together: \(\frac{2}{x} \times \frac{3}{2 x} = \frac{2 \times 3}{x \times 2x}\). Now, simplify the numerator and denominator: \(\frac{2 \times 3}{x \times 2x} = \frac{6}{2x^2}\). Finally, simplify: \(\frac{6}{2x^2} = \frac{3}{x^2}\).

Key concepts

Common Denominators

Understanding common denominators is crucial when it comes to simplifying rational expressions, especially when dealing with fractions that need to be added or subtracted. A common denominator refers to a shared multiple of the denominators of two or more fractions. Finding a common denominator allows fractions to be combined because it puts them on an equal footing.

To find the common denominator, identify the least common multiple (LCM) of the original denominators. For instance, if you have denominators of 3 and 4, the LCM is 12, so 12 would be the common denominator. When working with variables, such as \( x \) and \( 2x \) in our exercise, remember that \( 2x \) is already a multiple of \( x \) and thus can be used directly as the common denominator.

Converting fractions to have this shared base allows you to perform addition or subtraction seamlessly, which is the initial step in simplifying the expressions in our examples.

Subtracting Fractions

When subtracting fractions, the first step is always to ensure they have a common denominator. Once this is achieved, subtraction becomes straightforward: simply subtract the numerators (the top parts of the fractions) while keeping the denominator the same.

In the example \( \frac{2}{x}-\frac{3}{2x} \) we started by converting \( \frac{2}{x} \) to an equivalent fraction with the common denominator of \( 2x \) resulting in \( \frac{4}{2x} - \frac{3}{2x} \) (as \( 2\cdot 2 = 4 \) and \( x\cdot 2 = 2x \) respectively). The numerators were then subtracted to give us \( \frac{1}{2x} \) after simplification. Remember, the denominators always stay unchanged during the subtraction.

Adding Fractions

Much like subtraction, adding fractions also requires a common denominator. Once the common denominator is established, add the numerators only, keeping the common denominator intact.

Looking at our second example, \( \frac{2}{x}+\frac{3}{2x} \) we first ensure that both fractions have the common denominator of \( 2x \) which gives us \( \frac{4}{2x}+\frac{3}{2x} \) (using the same conversion as in the subtraction example). Adding the numerators—\( 4+3 \)—we get \( \frac{7}{2x} \) which is the simplified version of the original expression. Keep in mind, fractions with common denominators can be added directly without altering the denominator.

Multiplying Fractions

When it comes to multiplying fractions, the process is more straightforward than with addition or subtraction, because you don't need to find a common denominator first. Instead, simply multiply the numerators together to get a new numerator and multiply the denominators together for a new denominator.

In the exercise \( \frac{2}{x} \times \frac{3}{2x} \) multiplication leads us to \( \frac{2\times 3}{x \times 2x} \) which simplifies to \( \frac{6}{2x^2} \) and then further to \( \frac{3}{x^2} \) upon dividing both numerator and denominator by 2. This simplification is vital to ensure the expression is in its most reduced form, making it easier to work with in equations or further calculations.