https://file.aiquickdraw.com/mathbot/file/1725330783834q3g5wmco.jpg

Solve the equation. Check extraneous solutions for rational equations. Write answer in interval notation for rational inequalities 

The image contains the equation:

\[
\frac{4}{x} + \frac{1}{3x} = 9
\]

### Step 1: Find a common denominator

The common denominator between \(\frac{4}{x}\) and \(\frac{1}{3x}\) is \(3x\). So, rewrite the equation:

\[
\frac{4 \cdot 3}{3x} + \frac{1}{3x} = 9
\]

This simplifies to:

\[
\frac{12 + 1}{3x} = 9
\]

\[
\frac{13}{3x} = 9
\]

### Step 2: Solve for \(x\)

Multiply both sides by \(3x\) to eliminate the fraction:

\[
13 = 27x
\]

Then, solve for \(x\):

\[
x = \frac{13}{27}
\]

### Step 3: Check for Extraneous Solutions

Substitute \(x = \frac{13}{27}\) back into the original equation to check if it satisfies the equation:

\[
\frac{4}{\frac{13}{27}} + \frac{1}{3 \cdot \frac{13}{27}} = 9
\]

Simplify each term:

\[
\frac{4 \cdot 27}{13} + \frac{27}{3 \cdot 13} = 9
\]

\[
\frac{108}{13} + \frac{9}{13} = 9
\]

\[
\frac{117}{13} = 9
\]

Since \(\frac{117}{13} = 9\) is true, the solution \(x = \frac{13}{27}\) is valid.

### Interval Notation for Rational Inequalities

This particular equation does not directly involve inequalities, but if you were to consider the equation as an inequality, the solution would involve determining where the expression is less than or greater than 9.

For the solution to \( \frac{13}{3x} = 9 \), solving for \(x\) gives \(x = \frac{13}{27}\). This solution is valid for \(x > 0\), since \(x\) cannot be zero or negative (as the denominator would be zero or undefined).

### Final Answer:

The solution is \( x = \frac{13}{27} \).

**Interval Notation:**
If considering the solution in terms of inequalities, and assuming the inequality involves \( \geq 9 \), the solution would be in the interval:

\[ x \in \left(\frac{13}{27}, \infty \right) \]

Do you have any other questions or need further clarification?

Here are some related questions you might find helpful:
1. What are extraneous solutions and how do they arise?
2. How do you solve rational inequalities?
3. What is the significance of interval notation in inequalities?
4. How can you determine the domain of a rational function?
5. What are the steps to simplify complex rational expressions?

**Tip:** When solving rational equations or inequalities, always check for any restrictions on the variable, such as values that would make the denominator zero.

Learn how to solve rational equations, check for extraneous solutions, and express solutions in interval notation. This problem covers key concepts in equations, rational equations, inequalities, and interval notation. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation