https://file.aiquickdraw.com/mathbot/file/17289413454967k5fqfb5.jpg

Let's go through each inequality step-by-step to solve them.

### Problem 1:
\[
4x^3 - 125 \geq 0
\]
Factorize and solve for \( x \):
1. Rewrite \(4x^3 - 125\) as a difference of cubes: \((2x)^3 - 5^3\).
2. Use the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\):
   \[
   4x^3 - 125 = (2x - 5)(4x^2 + 10x + 25)
   \]
3. Set \(2x - 5 \geq 0\), so \(x \geq \frac{5}{2}\) and solve the quadratic part.

### Problem 2:
\[
\sqrt{2x + 1} < 3 + \sqrt{x + 2}
\]
1. Square both sides to eliminate the square roots (ensure conditions for non-negative values under square roots).
2. Simplify and solve for \( x \).

### Problem 3:
\[
\sqrt[3]{x^2 + 8} - 2 < x
\]
1. Rearrange: \(\sqrt[3]{x^2 + 8} < x + 2\).
2. Cube both sides and solve the resulting inequality.

### Problem 4:
\[
x + 1 > \sqrt[3]{x^2 + 1}
\]
1. Cube both sides (keeping in mind that \(x + 1\) must be positive).
2. Solve for \( x \).

### Problem 5:
\[
\sqrt[3]{(x - 2)(x^2 - x + 3)} > x - 2
\]
1. Consider cases based on \( x - 2 \) and cube both sides.
2. Solve for \( x \) in each case.

### Problem 6:
\[
16x^4 < 25
\]
1. Divide both sides by 16: \(x^4 < \frac{25}{16}\).
2. Take the fourth root of both sides and solve for \( x \).

### Problem 7:
\[
\sqrt{x - 1} - \sqrt[3]{x - 2} > 1
\]
1. Rearrange: \(\sqrt{x - 1} > 1 + \sqrt[3]{x - 2}\).
2. Square both sides and solve.

### Problem 8:
\[
\sqrt[3]{x^3 - 3x^2 + x + 1} \geq x - 1
\]
1. Cube both sides and simplify.
2. Solve for \( x \).

---

Would you like detailed solutions for each problem? Here are a few related questions:

1. How do we solve inequalities involving square roots?
2. What is the process for handling cube roots in inequalities?
3. How can we factor differences of cubes?
4. How do we handle fourth roots in inequalities?
5. What are the steps for solving inequalities involving multiple radical expressions?

**Tip:** When dealing with inequalities involving roots, remember to check the domain carefully to ensure all terms under the roots are valid.

Solve the following inequalities in R:
1) 4x^3 - 125 ≥ 0
2) √(2x + 1) < 3 + √(x + 2)
3) ∛(x^2 + 8) - 2 < x
4) x + 1 > ∛(x^2 + 1)
5) ∛((x - 2)(x^2 - x + 3)) > x - 2
6) 16x^4 < 25
7) √(x - 1) - ∛(x - 2) > 1
8) ∛(x^3 - 3x^2 + x + 1) ≥ x - 1

Learn how to solve radical and polynomial inequalities with step-by-step solutions. This problem covers algebraic concepts such as cube roots, square roots, and polynomials. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation