Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Determine whether each value of \(x\) is a solution of the equation. Equation $$ \frac{5}{2 x}-\frac{4}{x}=3 $$ Values (a) \(x=-\frac{1}{2}\) (b) \(x=4\) (c) \(x=0\) (d) \(x=\frac{1}{4}\)

Step-by-Step Solution

Answer

The solutions to the equation are \(x=-\frac{1}{2}\), \(x=4\), and \(x=\frac{1}{4}\). The value \(x=0\) is not a solution because it causes division by zero in the equation.

Step 1: Check Solution for \(x=-\frac{1}{2}\)

Substitute \(x=-\frac{1}{2}\) into the equation: \(\frac{5}{2(-\frac{1}{2})}-\frac{4}{-\frac{1}{2}}=3\). Simplify to check if the left-hand side equals to the right-hand side.

Step 2: Check Solution for \(x=4\)

Substitute \(x=4\) into the equation: \(\frac{5}{2(4)}-\frac{4}{4}=3\). Simplify to determine if the left-hand side equals to the right-hand side.

Step 3: Check Solution for \(x=0\)

Substitute \(x=0\) into the equation: \(\frac{5}{2(0)}-\frac{4}{0}=3\). This step immediately reveals that \(x=0\) is not a solution, because division by zero is undefined.

Step 4: Check Solution for \(x=\frac{1}{4}\)

Substitute \(x=\frac{1}{4}\) into the equation: \(\frac{5}{2(\frac{1}{4})}-\frac{4}{\frac{1}{4}}=3\). Simplify to check if the left-hand side equals to the right-hand side.

Key Concepts

Substitution Method

The substitution method involves taking potential solutions and plugging them into the equation to see if they satisfy it. To use this method effectively, you should carefully replace the variable with the given values, performing all necessary arithmetic operations to determine if both sides of the equation are equal.

For example, if we substitute a value like \(x = -\frac{1}{2}\) into an equation, we must replace every 'x' with \(-\frac{1}{2}\). This substitution allows us to transform the equation into a simple arithmetic problem:

• Begin by replacing \(x\) with \(-\frac{1}{2}\) in the equation.
• Calculate the resulting fractions and simplify as needed.
• If both sides of the equation equal the same value, then \(x = -\frac{1}{2}\) is a solution.

This method is direct and systematic, making it a reliable way to determine the validity of proposed solutions.

Rational Equations

Rational equations are equations that involve fractions, specifically those with polynomials in the numerator and/or denominator. Solving these equations typically involves finding common denominators or eliminating the fractions.

In the equation \( \frac{5}{2x} - \frac{4}{x} = 3 \), this involves:

• Identifying the denominators in the equation, which are \(2x\) and \(x\).
• Finding a common denominator, which would be \(2x\) in this context.
• Multiplying every term by the common denominator to eliminate the fractions.

Simplifying rational equations requires thoroughness in arithmetic operations and often involves verifying potential solutions to prevent errors.

Division by Zero

Division by zero is a mathematical no-go and is undefined, meaning you cannot divide by zero in any legitimate arithmetic operation. When solving rational equations, checking whether any value leads to division by zero is crucial.

In the example equation, when checking if \(x = 0\) is a solution, you would realize:

• Substituting \(x = 0\) makes the denominator zero in the fractions.
• This division by zero is not possible, hence \(x = 0\) cannot be a solution.

It is important to always check potential solutions in rational equations for this error to avoid incorrect conclusions.