Problem 10 - Algebra for College Students

Question

Find the solution set to each equation. $$\frac{5}{6 x}-\frac{1}{8 x}=\frac{17}{24}$$

Step-by-Step Solution

Answer

x = 1

Step 1: Find a common denominator

The given equation is \(\frac{5}{6x} - \frac{1}{8x} = \frac{17}{24}\). In order to simplify the left-hand side, it's important to find a common denominator for the fractions \(\frac{5}{6x}\) and \(\frac{1}{8x}\). The common denominator for 6x and 8x is 24x.

Step 2: Rewrite fractions with the common denominator

Rewrite \(\frac{5}{6x}\) and \(\frac{1}{8x}\) using the common denominator 24x. Multiply the numerator and denominator of each fraction by the necessary values to get to 24x:\(\frac{5}{6x} = \frac{5 \times 4}{6x \times 4} = \frac{20}{24x}\)\(\frac{1}{8x} = \frac{1 \times 3}{8x \times 3} = \frac{3}{24x}\)

Step 3: Subtract the fractions

Subtract the fractions \(\frac{20}{24x}\) and \(\frac{3}{24x}\):\(\frac{20}{24x} - \frac{3}{24x} = \frac{20 - 3}{24x} = \frac{17}{24x}\)

Step 4: Set up the equation

Now that the fractions on the left-hand side have been combined, substitute back into the original equation:\(\frac{17}{24x} = \frac{17}{24}\)

Step 5: Solve for x

To solve for x, multiply both sides of the equation by 24x to eliminate the fractions:\(\frac{17}{24x} \times 24x = \frac{17}{24} \times 24x\)This simplifies to:17 = 17xFinally, divide both sides by 17 to solve for x:x = 1

Step 6: Verify the solution

Plug the solution back into the original equation to make sure it's correct. Substituting x = 1:\(\frac{5}{6(1)} - \frac{1}{8(1)} = \frac{17}{24}\)This simplifies to:\(\frac{5}{6} - \frac{1}{8} = \frac{17}{24}\)Finding a common denominator for the fractions\(\frac{5}{6}\) and \(\frac{1}{8}\) (which is 24), the equation simplifies to true statement:\(\frac{20}{24} - \frac{3}{24} = \frac{17}{24}\)Since both sides of the equation are equal, the solution x = 1 is correct.

Key Concepts

common denominator

In rational equations, finding a common denominator is a crucial step. It helps to combine fractions and simplify the equation. Take, for example, the equation \(\frac{5}{6x} - \frac{1}{8x} = \frac{17}{24}\). Both fractions on the left share 'x' in the denominator, but the numerical parts (6 and 8) are different. To harmonize these, we need to find a shared multiple of 6 and 8, which in this case is 24. By rewriting each fraction to have this common denominator (24x), we make subtraction easier and more straightforward. This simplification involves multiplying each fraction's numerator and denominator by the necessary factors. Understanding common denominators ensures clarity in each step of solving rational equations.

fraction subtraction

Once fractions share the same denominator, subtracting them is straightforward. In the given problem \( \frac{20}{24x} - \frac{3}{24x} \), both fractions were rewritten to have the common denominator 24x. Subtract the numerators (20 and 3) while keeping the denominator unchanged. This results in \( \frac{17}{24x} \). This simplification is important because it reduces complex fractions into simpler forms that are much easier to handle. Subtracting fractions correctly ensures the equation remains balanced and set up for the next solving steps.

cross-multiplication

Cross-multiplication is used to eliminate denominators and simplify rational equations. When we set up the equation \( \frac{17}{24x} = \frac{17}{24} \), both sides have the fraction 17 over 24 times x. By multiplying both sides by 24x, the denominators cancel out. This operation simplifies the equation to 17 = 17x. Cross-multiplication is powerful because it transforms an equation with rational expressions into a simpler, solvable form. It's essential to keep equations balanced, operating equally on both sides to maintain the equation's integrity.

verify solution

Verification ensures that our solution is correct. After solving for x (x = 1), substitute it back into the original equation to check: \( \frac{5}{6 \times 1} - \frac{1}{8 \times 1} = \frac{17}{24} \). Simplifying each term, we get \( \frac{5}{6} - \frac{1}{8} = \frac{17}{24} \). Rewriting \( \frac{5}{6} \) and \( \frac{1}{8} \) using the common denominator 24, calculates as \( \frac{20}{24} - \frac{3}{24} = \frac{17}{24} \). Both sides match, confirming that x = 1 is indeed the correct solution. Verification steps help catch potential errors and ensure that the final answer solves the initial equation correctly.