Let's solve the inequality \(\frac{2r - 5}{3r} > -\frac{1}{3}\).

### Step 1: Clear the fractions by multiplying both sides by \(3r\).
Since \(3r\) is a variable expression, we need to consider the sign of \(r\) because it will affect the inequality sign. We will solve the inequality for two cases: \(r > 0\) and \(r < 0\).

#### Case 1: \(r > 0\)
When \(r > 0\), multiplying both sides by \(3r\) preserves the inequality sign:

\[
2r - 5 > -\frac{1}{3} \cdot 3r
\]

This simplifies to:

\[
2r - 5 > -r
\]

### Step 2: Solve for \(r\).
Add \(r\) to both sides:

\[
3r - 5 > 0
\]

Add 5 to both sides:

\[
3r > 5
\]

Divide by 3:

\[
r > \frac{5}{3}
\]

#### Case 2: \(r < 0\)
When \(r < 0\), multiplying both sides by \(3r\) reverses the inequality sign:

\[
2r - 5 < -r
\]

### Step 3: Solve for \(r\).
Add \(r\) to both sides:

\[
3r - 5 < 0
\]

Add 5 to both sides:

\[
3r < 5
\]

Divide by 3:

\[
r < \frac{5}{3}
\]

### Step 4: Combine the results.
From Case 1, we have \(r > \frac{5}{3}\).
From Case 2, we have \(r < \frac{5}{3}\).

Since these cases involve different intervals, the only solution that satisfies both cases is the union of the results. Therefore:

- For \(r > 0\), \(r > \frac{5}{3}\).
- For \(r < 0\), \(r < \frac{5}{3}\).

### Final Answer:
The solution is \(r > \frac{5}{3}\) when \(r > 0\), and \(r < \frac{5}{3}\) when \(r < 0\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you solve inequalities with multiple fractions?
2. Can inequalities involving variables in the denominator have no solutions?
3. What are the rules for reversing inequality signs when multiplying by negative numbers?
4. How can you check if a solution satisfies the original inequality?
5. What is the difference between strict inequalities and non-strict inequalities?

**Tip:** Always be cautious when multiplying or dividing inequalities by a variable expression, as the sign of the expression can change the direction of the inequality.

Learn how to solve the fractional inequality (2r - 5) / (3r) > -1/3 with detailed step-by-step solutions. Understand the method to clear fractions, handle inequalities involving variables in the denominator, and find solutions for different cases of the variable r. Suitable for Grades 10-12 students studying inequalities and algebraic manipulations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation