Problem 12 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{5}{2 x}-\frac{8}{9}=\frac{1}{18}-\frac{1}{3 x}$$

Step-by-Step Solution

Answer

The solution to the given rational equation is \(x = -54/28\), which further simplifies to \(x = -27/14\).

Step 1: Simplifying the equation

The first step is to simplify the equation by creating the least common multiple among the denominators. That would be \(18x\). Multiply each term of the equation by \(18x\) to eliminate the denominators.

Step 2: Distributing and combining like terms

After step 1, the equation becomes \(45x - 16x = x - 54\). Now combine like terms to simplify this equation into \(29x = x - 54\).

Step 3: Solving for x

Following step 2, you subtract x from both sides to isolate x term on one side. The equation now becomes \(28x = -54\). Finally, divide by 28 to solve for x to get \(x = -54/28\).

Key Concepts

Understanding Least Common Multiple

When working with rational equations, identifying the least common multiple (LCM) of the denominators is crucial. The LCM is the smallest number that each of the denominators can evenly divide into. Without finding the LCM, you cannot effectively clear the fractions from the equation.
To find the LCM of several denominators in a rational equation, you need to:

• Factor each denominator to its prime components.
• Identify the highest power of each factor among the fractions.
• Multiply these factors together to calculate the LCM.

In the example provided, the denominators are 2x, 9, 18, and 3x. The LCM of these is 18x, which is the smallest number into which all these denominators can divide evenly. By multiplying the entire equation by the LCM, 18x, you remove the fractions, making it easier to solve.

Steps in Solving Equations

Solving equations involves finding the values of variables that make the equation true. For rational equations, after eliminating fractions by using the LCM, the process becomes similar to solving regular linear equations. Here are steps to make it easier:

• After clearing fractions, distribute any multiplied factors that apply.
• Combine like terms - this means adding or subtracting terms with the same variable.
• Simplify the equation as much as possible.

In our problem, once you multiply through by 18x, the equation simplifies to 45x - 16x = x - 54. This simplification helps in the next step of isolating the variable to find the solution.

Isolating Variables Made Simple

Isolating the variable is a key step in finding the solution to an equation. This means you need to have the variable term by itself on one side of the equation. Here's how to achieve this:

• Move the terms containing the variable to one side of the equation.
• Move constant terms to the opposite side.
• Divide or multiply as needed to solve for the variable.

In the presented rational equation, after combining like terms, you reach the form 29x = x - 54. By subtracting x from both sides, you simplify it to 28x = -54.
Finally, divide every term by 28 to get the solution, leading to x = -54/28. This precise calculation helps ensure the accuracy of your solution, and by isolating x, you see clearly its value in the equation.