Problem 89 - Intermediate Algebra

Question

For each expression in part (a), perform the indicated operations and then simplify, if possible. Solve equation in part (b) and check the result. a. \(\frac{11}{12}-\frac{3}{2 x}+\frac{4}{x}\) b. \(\frac{11}{12}-\frac{3}{2 x}=\frac{4}{x}\)

Step-by-Step Solution

Answer

a. \( \frac{11x + 30}{12x} \); b. \( x = 6 \).

Step 1: Find a Common Denominator for Part (a)

To combine the fractions in part (a) of the expression \( \frac{11}{12} - \frac{3}{2x} + \frac{4}{x} \), we need to find a common denominator. The denominators are 12, 2x, and x. The least common denominator (LCD) for these is 12x.

Step 2: Convert Each Fraction to the Common Denominator

Transform each fraction to have the common denominator, 12x:- For \( \frac{11}{12} \): Multiply both the numerator and the denominator by x to get \( \frac{11x}{12x} \).- For \( \frac{3}{2x} \): Multiply both the numerator and the denominator by 6 to get \( \frac{18}{12x} \).- For \( \frac{4}{x} \): Multiply both the numerator and the denominator by 12 to get \( \frac{48}{12x} \).

Step 3: Combine the Fractions

Now that all terms are over the same denominator, we can combine them:\[ \frac{11x}{12x} - \frac{18}{12x} + \frac{48}{12x} = \frac{11x - 18 + 48}{12x} = \frac{11x + 30}{12x} \].Thus, the simplified form of the expression is \( \frac{11x + 30}{12x} \).

Step 4: Solve Equation in Part (b)

The equation \( \frac{11}{12} - \frac{3}{2x} = \frac{4}{x} \) should be solved by clearing the fractions. Multiply each term by the common denominator 12x to eliminate the denominators:\( 11x - 18 = 48 \).

Step 5: Isolate the Variable x

Rearrange the equation from Step 4 to solve for x:1. \( 11x - 18 = 48 \).2. Add 18 to both sides to get: \( 11x = 66 \).3. Divide both sides by 11 to isolate x: \( x = \frac{66}{11} \).4. Simplify to find: \( x = 6 \).

Step 6: Verify the Solution for x

Substitute \( x = 6 \) back into the original equation to check:- The original equation: \( \frac{11}{12} - \frac{3}{2(6)} = \frac{4}{6} \).- Simplify: \( \frac{11}{12} - \frac{3}{12} = \frac{2}{3} \).- Simplify further: \( \frac{8}{12} = \frac{2}{3} \) is true.Thus, the solution \( x = 6 \) is verified.

Key Concepts

Common Denominator

A common denominator is essential when you have multiple fractions that you need to add or subtract. Simply put, it's a shared multiple of the denominators in a set of fractions. Finding a common denominator allows you to rewrite the fractions so they can be easily combined. With our exercise, we needed a common denominator for the fractions in the expression \( \frac{11}{12} - \frac{3}{2x} + \frac{4}{x} \). The denominators here are 12, 2x, and x.

By finding a single denominator that each original denominator can divide into with no remainder, we can effectively combine our fractions. This requires examining each denominator to find their

• Least Common Multiple (LCM)
• shared product

In such cases, the common denominator truly simplifies our math work! For this set, the common denominator turns out to be 12x.

Simplifying Fractions

Once you have a common denominator, simplifying fractions becomes very straightforward. You can represent each fraction's numerator with respect to this common denominator, allowing you to simplify the combined equation more easily. In our problem, we adjusted each term to have the denominator of 12x.

The conversion worked like this:

• For \( \frac{11}{12} \): multiply both numerator and denominator by x to get \( \frac{11x}{12x} \).
• For \( \frac{3}{2x} \): multiply both parts by 6 to reach \( \frac{18}{12x} \).
• And \( \frac{4}{x} \): both parts multiplied by 12 produce \( \frac{48}{12x} \).

With all fractions sharing the same denominator, it's a simple task to add or subtract the fractions by focusing only on the numerators. Thus, simplifying becomes a matter of basic arithmetic: \( \frac{11x + 30}{12x} \) is the simplified expression.

Solving Equations

Solving equations that involve fractions can feel tricky, but they are easier than they might appear. Begin by eliminating fractions, often by multiplying every term by the Least Common Denominator (LCD) to streamline the equation. For part (b) of our problem, this involved converting the equation \( \frac{11}{12} - \frac{3}{2x} = \frac{4}{x} \) to eliminate the fractions.

By multiplying every term by 12x, we create:

• \( 11x - 18 = 48 \)

This linear equation is simple compared to dealing with three individual fractions. With fractions gone, solve for x with basic algebra.

Least Common Denominator (LCD)

The least common denominator is a powerful tool! It's the smallest multiple that is common among the original denominators in your fractions. When faced with different denominators—like our 12, 2x, and x—the least common denominator (LCD) is what you'll use to rewrite fractions neatly.

The LCD helps in converting different denominators into a unified approach, making additions and subtractions easy and accurate. Here’s how it's determined:

• Find the greatest power of each prime number appearing in the factorization of each denominator.
• For 12 (which equals \(2^2 \cdot 3\)), 2x, and x (both with factors of x), you focus on all involved components.
• The LCD is 12x, the smallest number each divides into without leaving a fraction behind.

Through LCD, any mix of fractions can be made ready for addition, subtraction, or comparison, avoiding any convoluted math clutter!