prove that $$ \frac{n}{n^2-5n+4}\ge\frac{1}{n} $$for n$$ \ge $$5

To prove the inequality 

\[
\frac{n}{n^2 - 5n + 4} \geq \frac{1}{n}
\]

for $ n \geq 5 $, let's begin by simplifying the inequality.

### Step 1: Multiply both sides by $ n(n^2 - 5n + 4) $
Since $ n > 0 $ and $ n^2 - 5n + 4 > 0 $ for $ n \geq 5 $, multiplying both sides by these terms won't change the direction of the inequality.

\[
n^2 \geq n^2 - 5n + 4
\]

### Step 2: Simplify the inequality
Subtract $ n^2 $ from both sides:

\[
0 \geq -5n + 4
\]

### Step 3: Solve for $ n $
Add $ 5n $ to both sides:

\[
5n \geq 4
\]

Now, divide both sides by 5:

\[
n \geq \frac{4}{5}
\]

### Step 4: Compare the result with the given condition
The inequality $ n \geq \frac{4}{5} $ is always true for $ n \geq 5 $. Since $ n \geq 5 $ is our initial condition, the inequality holds for all $ n \geq 5 $.

### Conclusion
The given inequality 

\[
\frac{n}{n^2 - 5n + 4} \geq \frac{1}{n}
\]

is true for $ n \geq 5 $.

---

Would you like more details on any of these steps? Here are some related questions that might interest you:

1. How do you analyze the behavior of rational functions as $ n $ approaches infinity?
2. Can inequalities be proven using calculus methods?
3. How does one factor quadratic expressions like $ n^2 - 5n + 4 $?
4. What other methods can be used to prove inequalities?
5. How do constraints on $ n $ affect the validity of an inequality?

**Tip:** When proving inequalities, always check if the operations performed (like multiplying or dividing by a term) affect the direction of the inequality.

Learn how to prove the inequality n / (n^2 - 5n + 4) >= 1 / n for n >= 5 with a detailed step-by-step solution. This problem involves inequalities and rational functions, suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation