Solve each rational inequality then show its interval of values.
1. 2
x+5
<
5
11

To solve the rational inequality \(\frac{2}{x+5} < \frac{5}{11}\), follow these steps:

### Step 1: Set up the inequality
Given:
\[
\frac{2}{x+5} < \frac{5}{11}
\]

### Step 2: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[
2 \times 11 < 5 \times (x + 5)
\]
\[
22 < 5x + 25
\]

### Step 3: Isolate \(x\)
Subtract 25 from both sides:
\[
22 - 25 < 5x
\]
\[
-3 < 5x
\]

Divide by 5:
\[
-\frac{3}{5} < x
\]
Or, equivalently:
\[
x > -\frac{3}{5}
\]

### Step 4: Determine the interval of values
The solution to the inequality is:
\[
x > -\frac{3}{5}
\]
This means \(x\) can take any value greater than \(-\frac{3}{5}\).

### Interval notation
In interval notation, the solution is:
\[
\left(-\frac{3}{5}, \infty \right)
\]

### Summary
The solution to the inequality is \(x > -\frac{3}{5}\), or \(\left(-\frac{3}{5}, \infty \right)\).

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Would you like further details or have any questions?

Here are some related questions you can explore:
1. Solve the inequality \(\frac{3}{x-4} \geq \frac{1}{2}\).
2. How do you solve inequalities with rational expressions?
3. What are the steps to cross-multiply correctly in inequalities?
4. Explain how interval notation works with inequalities.
5. Solve the inequality \(\frac{5}{x+2} > \frac{3}{7}\).
6. What are the differences between solving linear inequalities and rational inequalities?
7. Discuss the importance of checking critical points when solving rational inequalities.
8. How do you represent the solution of an inequality on a number line?

**Tip:** When solving rational inequalities, always consider the domain of the expression to avoid dividing by zero.

Learn how to solve the rational inequality 2/(x+5) < 5/11 step-by-step. Understand the concepts of inequalities, rational expressions, and interval notation. Suitable for high school students.

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