https://file.aiquickdraw.com/mathbot/file/1725484459225r0yxd75j.jpg

The inequality in the image is:

\[
x^2 + 7x \geq 4
\]

Let's solve this inequality.

### Step 1: Move all terms to one side of the inequality.
Subtract 4 from both sides:

\[
x^2 + 7x - 4 \geq 0
\]

### Step 2: Solve the corresponding quadratic equation.
To find the critical points, solve the quadratic equation:

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

Use the quadratic formula:

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

where \(a = 1\), \(b = 7\), and \(c = -4\).

\[
x = \frac{-7 \pm \sqrt{7^2 - 4(1)(-4)}}{2(1)}
\]
\[
x = \frac{-7 \pm \sqrt{49 + 16}}{2}
\]
\[
x = \frac{-7 \pm \sqrt{65}}{2}
\]

So the solutions are:

\[
x = \frac{-7 + \sqrt{65}}{2} \quad \text{and} \quad x = \frac{-7 - \sqrt{65}}{2}
\]

### Step 3: Analyze the intervals.
The critical points divide the number line into three intervals. Test the sign of the quadratic expression \(x^2 + 7x - 4\) in each interval to determine where the inequality holds.

1. \(x < \frac{-7 - \sqrt{65}}{2}\)
2. \(\frac{-7 - \sqrt{65}}{2} \leq x \leq \frac{-7 + \sqrt{65}}{2}\)
3. \(x > \frac{-7 + \sqrt{65}}{2}\)

### Step 4: Conclusion
- The quadratic expression is greater than or equal to zero in intervals where the test values yield positive results.
- Therefore, the solution to the inequality is:

\[
x \in \left(-\infty, \frac{-7 - \sqrt{65}}{2}\right] \cup \left[\frac{-7 + \sqrt{65}}{2}, \infty\right)
\]

Would you like a more detailed explanation on any step or have any questions?

---

Here are 5 related questions that you may find helpful:

1. How do you determine the sign of a quadratic expression in different intervals?
2. What is the significance of the discriminant in the quadratic formula?
3. Can this inequality be solved using a different method?
4. How would the solution change if the inequality were strict (i.e., \(>\) instead of \(\geq\))?
5. What are the applications of solving quadratic inequalities?

**Tip:** Always check the sign of a quadratic expression by testing values in each interval defined by the roots of the equation. This helps in accurately determining the solution set for inequalities.

Learn how to solve the quadratic inequality x^2 + 7x ≥ 4 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and analyzing the sign of the expression across different intervals. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation