x*2+7x/x-4 < 0 solve this rational inequality

To solve the rational inequality:

\[
\frac{x^2 + 7x}{x - 4} < 0
\]

### Step 1: Factor the numerator.
Factor the numerator \(x^2 + 7x\):

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

So, the inequality becomes:

\[
\frac{x(x + 7)}{x - 4} < 0
\]

### Step 2: Find the critical points.
The critical points are the values of \(x\) that make the numerator or denominator equal to zero. Set the numerator and denominator equal to zero:

1. \(x = 0\) from \(x = 0\)
2. \(x + 7 = 0 \Rightarrow x = -7\)
3. \(x - 4 = 0 \Rightarrow x = 4\)

So, the critical points are \(x = -7\), \(x = 0\), and \(x = 4\).

### Step 3: Determine the sign of the expression in each interval.
The critical points divide the real number line into four intervals:

1. \( (-\infty, -7) \)
2. \( (-7, 0) \)
3. \( (0, 4) \)
4. \( (4, \infty) \)

We will test the sign of the expression \(\frac{x(x + 7)}{x - 4}\) in each interval.

1. **Interval \( (-\infty, -7) \):** Pick \(x = -8\)
   \[
   \frac{-8(-8 + 7)}{-8 - 4} = \frac{-8(-1)}{-12} = \frac{8}{12} > 0
   \]
   The expression is positive in this interval.

2. **Interval \( (-7, 0) \):** Pick \(x = -1\)
   \[
   \frac{-1(-1 + 7)}{-1 - 4} = \frac{-1(6)}{-5} = \frac{-6}{-5} > 0
   \]
   The expression is positive in this interval.

3. **Interval \( (0, 4) \):** Pick \(x = 2\)
   \[
   \frac{2(2 + 7)}{2 - 4} = \frac{2(9)}{-2} = \frac{18}{-2} = -9 < 0
   \]
   The expression is negative in this interval.

4. **Interval \( (4, \infty) \):** Pick \(x = 5\)
   \[
   \frac{5(5 + 7)}{5 - 4} = \frac{5(12)}{1} = 60 > 0
   \]
   The expression is positive in this interval.

### Step 4: Write the solution.
We are looking for where the expression is less than zero, which happens in the interval where the expression is negative.

The solution is:

\[
x \in (0, 4)
\]

### Step 5: Exclude critical points where the expression is undefined.
At \(x = 4\), the denominator is zero, so this point is excluded from the solution.

The final solution is:

\[
x \in (0, 4)
\]

---

Would you like further clarification on any step? Here are some related questions:

1. How can we graphically represent the solution of this inequality?
2. What if the inequality were \(\leq\) instead of \(<\)?
3. How do you solve inequalities where the numerator has higher degree than the denominator?
4. What if the inequality were \(\frac{x^2 + 7x}{x - 4} > 0\)?
5. How do you handle rational inequalities with multiple factors in the denominator?

**Tip:** When solving rational inequalities, pay attention to where the expression is undefined, as this can change the solution intervals.

Learn how to solve the rational inequality (x^2 + 7x) / (x - 4) < 0 with a detailed step-by-step solution. This problem covers key algebraic concepts, including rational inequalities, factoring of quadratics, and critical point analysis. Suitable for high school students.

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