Problem 29 - A Graphical Approach to College Algebra

Question

Find all complex solutions for each equation by hand. Do not use a calculator. $$9 x^{-1}+4(6 x-3)^{-1}=2(6 x-3)^{-1}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{27}{56}\).

Step 1: Identify Common Denominator

To solve the equation \(9 x^{-1}+4(6 x-3)^{-1}=2(6 x-3)^{-1}\), first rewrite the equation in terms of fractions: \(\frac{9}{x} + \frac{4}{6x-3} = \frac{2}{6x-3}\). The common denominator of the fractions \(\frac{9}{x}\) and \(\frac{4}{6x-3}\) is \(x(6x-3)\).

Step 2: Rewrite Equation with Common Denominator

Write the fractions over the common denominator: \(\frac{9(6x-3) + 4x}{x(6x-3)} = \frac{2x}{x(6x-3)}\). Now both sides have the same denominator, which allows us to focus on the numerators.

Step 3: Simplify the Numerators

Simplify the numerators to clear the fraction: \(9(6x-3) + 4x = 2x\). Distribute the 9: \(54x - 27 + 4x = 2x\). Combine like terms: \(58x - 27 = 2x\).

Step 4: Isolate Variable Terms

Subtract \(2x\) from both sides to isolate the variable terms: \(58x - 2x = 27\). Simplify: \(56x = 27\).

Step 5: Solve for x

Divide both sides by 56 to solve for \(x\): \(x = \frac{27}{56}\). This gives the solution \(x = \frac{27}{56}\). Since the problem deals with complex solutions and no further complex numbers are involved in the process, this is our answer.

Key Concepts

Common Denominator

When dealing with equations involving fractions, finding a common denominator is usually the first step.This makes it easier to work with the equation as you can combine or compare the fractions successfully.In the given equation, you have the fractions \(\frac{9}{x}\) and \(\frac{4}{6x-3}\).
The common denominator will be the product of the individual denominators, which is \(x(6x-3)\).

• Write each term of the equation with this common denominator.
• Ensure all fractions are expressed over \(x(6x-3)\) to make calculations simpler.

Understanding how to find a common denominator not only helps in simplifying fractions but also sets the stage for further simplification steps in solving equations.

Fraction Simplification

Simplifying fractions involves reducing the terms in the numerator and denominator.Once the common denominator is established, rewrite each fraction using it and then focus on simplifying the numerators.In this case, we rewrite \[\frac{9(6x-3) + 4x}{x(6x-3)} = \frac{2x}{x(6x-3)}\]into a simpler form.

• First, distribute the numbers in the numerator to eliminate parentheses.
• For example, distribute 9 in \(9(6x-3)\) to get \(54x - 27\).
• Combine any like terms; here, \(54x\) and \(4x\) become \(58x\).

Every bit you simplify removes complexity and helps you move closer to finding the solution.After simplification, equations are easier to manipulate towards the solution.

Isolate Variable

The next step after simplifying fractions is to isolate the variable, in this case, \(x\).Once the numerators are simplified, set them equal to each other as you will have eliminated the denominator through previous steps.You then have a much simpler equation to work with, which is \(58x - 27 = 2x\).

• Subtract \(2x\) from both sides: \(58x - 2x = 27\).
• This results in \(56x = 27\).

At this point, isolating the variable is straightforward.Divide each side by 56 to solve for \(x\): \(x = \frac{27}{56}\).This is the solution to the equation.The goal is always to have the variable on one side of the equation by itself, enabling you to solve for its value easily.