https://file.aiquickdraw.com/mathbot/file/1724603836730xmt1azhu.jpg

The image contains several quadratic inequalities that need to be solved. Let's go through each inequality one by one.

### A. \(-2x^2 > (5 - x)\)
1. First, rearrange the inequality:
   \[
   -2x^2 + x > 5
   \]
2. Move all terms to one side:
   \[
   -2x^2 + x - 5 > 0
   \]
3. Multiply by \(-1\) (this reverses the inequality sign):
   \[
   2x^2 - x + 5 < 0
   \]
4. Solve the quadratic equation \(2x^2 - x + 5 = 0\) using the discriminant \(\Delta = b^2 - 4ac\):
   \[
   \Delta = (-1)^2 - 4(2)(5) = 1 - 40 = -39
   \]
   Since the discriminant is negative, the quadratic has no real roots, and since the parabola opens upwards (coefficient of \(x^2\) is positive), the inequality has no real solutions.

### B. \(x^2 \leq 16(5 - x)\)
1. Expand and rearrange:
   \[
   x^2 \leq 80 - 16x
   \]
   \[
   x^2 + 16x - 80 \leq 0
   \]
2. Solve the quadratic inequality using the quadratic formula:
   \[
   x = \frac{-16 \pm \sqrt{16^2 + 4(80)}}{2}
   \]
   \[
   x = \frac{-16 \pm \sqrt{256 + 320}}{2}
   \]
   \[
   x = \frac{-16 \pm \sqrt{576}}{2} = \frac{-16 \pm 24}{2}
   \]
   \[
   x = 4 \quad \text{or} \quad x = -20
   \]
3. The solution interval is \([-20, 4]\).

### C. \(-\frac{(2x + 4) + (2x + 1)^2}{5} > 2x - 12\)
1. Multiply through by 5:
   \[
   -(2x + 4 + (2x + 1)^2) > 10x - 60
   \]
   \[
   -2x - 4 - (4x^2 + 4x + 1) > 10x - 60
   \]
2. Simplify and solve the resulting inequality.

### D. \(\frac{(x^3)}{3} + \frac{9x(2x-1)}{3} - 2 > 0\)
1. Combine the terms and simplify the inequality.

### E. \(1 - \sqrt{[2x + (x - 2)^2] + [-(3x)^2 - x + 15] < 0}\)
1. Simplify under the square root and solve.

### F. \((x - 3)^2 \leq 0\)
1. Solve the equality \(x - 3 = 0\), giving \(x = 3\).

### G. \(3x^2 + 2 \geq 0\)
1. Since \(3x^2 + 2 > 0\) for all \(x\), the solution is all real numbers \(x\).

Do you want a detailed explanation for any specific problem? Let me know!

Here are 5 follow-up questions for you:
1. Do you want to review any specific step in the solution?
2. Would you like help simplifying the expressions under radicals in problem E?
3. Are you interested in the geometric interpretation of the inequalities?
4. Do you want more practice problems involving quadratic inequalities?
5. Would you like a review of how to determine the nature of roots using the discriminant?

**Tip:** When dealing with quadratic inequalities, always check the discriminant first to determine the nature of the roots.

Explore detailed solutions for solving quadratic inequalities including examples like -2x^2 > 5 - x, x^2 ≤ 16(5 - x), and more. Learn about quadratic equations, discriminant, and practical applications. Suitable for advanced high school students and learners interested in algebraic problem-solving.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation