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The inequality given is:

\[
3x^2 - 7x - 4 < 0
\]

We need to solve this quadratic inequality and express the solution in interval notation. Let's follow these steps:

### Step 1: Solve the quadratic equation
First, solve the corresponding quadratic equation:

\[
3x^2 - 7x - 4 = 0
\]

We can solve this using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(3x^2 - 7x - 4 = 0\), the coefficients are:
- \(a = 3\)
- \(b = -7\)
- \(c = -4\)

Let's compute the solutions using the quadratic formula.The solutions to the quadratic equation \(3x^2 - 7x - 4 = 0\) are:

\[
x_1 = 2.81 \quad \text{(approximately)} 
\]
\[
x_2 = -0.47 \quad \text{(approximately)} 
\]

### Step 2: Analyze the inequality
Next, we analyze the inequality \(3x^2 - 7x - 4 < 0\). The quadratic expression represents a parabola that opens upwards (since the coefficient of \(x^2\) is positive). The expression will be negative between the two roots.

Thus, the solution to the inequality is:

\[
-0.47 < x < 2.81
\]

### Step 3: Express the solution in interval notation
The solution in interval notation is:

\[
(-0.47, 2.81)
\]

Would you like more details or clarifications? Here are five related questions to deepen your understanding:

1. How do we handle quadratic inequalities when the parabola opens downwards?
2. How does the discriminant help in solving quadratic equations?
3. What would change in the solution if the inequality were \( \geq 0 \) instead of \( < 0 \)?
4. How do you verify that a solution to a quadratic inequality is correct?
5. Can quadratic inequalities have solutions over an infinite range?

**Tip:** Always check the sign of the leading coefficient in a quadratic expression to determine the direction of the parabola.

Find all values of x such that 3x^2 - 7x - 4 < 0. Enter your answer using interval notation.

Learn how to solve the quadratic inequality 3x^2 - 7x - 4 < 0 by using the quadratic formula and analyzing the roots. This problem covers key algebraic concepts and is suitable for students in Grades 9-11.

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