1/(7+x)< 7/(7x+x^2) Find the solution set of the given inequality by testing signs between potential sign-changing points. (Enter your answer using interval notation.) Show it on a numberline 

To solve the inequality \(\frac{1}{7 + x} < \frac{7}{7x + x^2}\), we'll follow these steps:

### Step 1: Simplify the inequality
We'll first eliminate the fractions by cross-multiplying. But, we must consider the sign of both sides to ensure that the direction of the inequality is correct:

\[
\frac{1}{7 + x} < \frac{7}{7x + x^2}
\]

Multiply both sides by \((7 + x)(7x + x^2)\), keeping in mind that this expression could be positive or negative depending on the values of \(x\):

\[
(7x + x^2) < 7(7 + x)
\]

### Step 2: Expand and simplify
Now, expand both sides:

\[
7x + x^2 < 49 + 7x
\]

Subtract \(7x\) from both sides:

\[
x^2 < 49
\]

### Step 3: Solve the quadratic inequality
This inequality can be rewritten as:

\[
x^2 - 49 < 0
\]

This is a difference of squares:

\[
(x - 7)(x + 7) < 0
\]

### Step 4: Determine intervals
The solutions to the equation \((x - 7)(x + 7) = 0\) are \(x = 7\) and \(x = -7\).

We now determine the sign of the product \((x - 7)(x + 7)\) in the intervals:

1. \(x < -7\)
2. \(-7 < x < 7\)
3. \(x > 7\)

### Step 5: Test the signs in each interval
- For \(x < -7\) (e.g., \(x = -8\)): \((x - 7)\) is negative, \((x + 7)\) is negative, and their product is positive.
- For \(-7 < x < 7\) (e.g., \(x = 0\)): \((x - 7)\) is negative, \((x + 7)\) is positive, and their product is negative.
- For \(x > 7\) (e.g., \(x = 8\)): \((x - 7)\) is positive, \((x + 7)\) is positive, and their product is positive.

The inequality \((x - 7)(x + 7) < 0\) holds when \(-7 < x < 7\).

### Step 6: Consider the domain restrictions
We must also consider where the original expressions are undefined, particularly where \(7 + x = 0\) or \(7x + x^2 = 0\):

- \(7 + x = 0\) implies \(x = -7\)
- \(7x + x^2 = x(7 + x) = 0\) implies \(x = 0\) or \(x = -7\)

So, the expression is undefined at \(x = -7\) and \(x = 0\).

### Step 7: Combine the results
Considering the domain and the sign analysis, the solution set is:

\[
(-7, 0) \cup (0, 7)
\]

This solution excludes the points where the expression is undefined.

### Step 8: Show the solution on a number line
- Draw a number line.
- Place open circles at \(x = -7\), \(x = 0\), and \(x = 7\) to indicate these points are not included in the solution set.
- Shade the interval \((-7, 0)\) and \((0, 7)\).

Would you like a visual representation or a more detailed explanation?

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**Relative Questions:**

1. How do you determine the intervals where the inequality is satisfied?
2. What is the significance of the sign of the product in interval testing?
3. How do domain restrictions impact the solution of inequalities?
4. What happens if one of the terms in the inequality is a complex expression?
5. How would the solution change if the inequality was non-strict (i.e., \(\leq\) or \(\geq\))?

**Tip:** Always check the domain of the original inequality before concluding the solution, especially when dealing with fractions or square roots.

Learn how to solve the inequality 1/(7+x) < 7/(7x+x^2) step-by-step with detailed explanations. This problem covers rational expressions, quadratic equations, and interval notation. Suitable for high school students in Grades 10-12.

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