https://file.aiquickdraw.com/mathbot/file/1725925729670vtnn1uex.jpg

The image contains three inequality problems. Let's solve each of them step-by-step:

### 1. Problem 1
\[
\frac{3x - 15}{x} \leq 5 - x
\]

**Solution:**

1. Clear the fraction by multiplying both sides by \(x\) (assuming \(x \neq 0\)):
   \[
   3x - 15 \leq x(5 - x)
   \]

2. Distribute and simplify:
   \[
   3x - 15 \leq 5x - x^2
   \]

3. Rearrange the inequality to form a quadratic equation:
   \[
   x^2 - 2x - 15 \geq 0
   \]

4. Factor the quadratic:
   \[
   (x - 5)(x + 3) \geq 0
   \]

5. Determine the critical points: \(x = 5\) and \(x = -3\). Analyze the intervals: 
   \(-\infty, -3\), \((-3, 5)\), and \( (5, \infty)\).

6. Test values from each interval to find where the inequality holds:
   \[
   \text{The solution is: } x \in (-\infty, -3] \cup [5, \infty).
   \]

### 2. Problem 2
\[
\frac{4t + 5}{t} > 3
\]

**Solution:**

1. Clear the fraction by multiplying both sides by \(t\) (assuming \(t \neq 0\)):
   \[
   4t + 5 > 3t
   \]

2. Rearrange the inequality:
   \[
   t > -5
   \]

3. Determine the valid range of \(t\):
   \[
   \text{The solution is: } t > 0 \text{ and } t > -5 \implies t > 0.
   \]

### 3. Problem 3
\[
\frac{2x}{x+1} \geq 1
\]

**Solution:**

1. Move 1 to the left side:
   \[
   \frac{2x}{x+1} - 1 \geq 0
   \]

2. Combine fractions:
   \[
   \frac{2x - (x + 1)}{x+1} \geq 0 \implies \frac{x - 1}{x+1} \geq 0
   \]

3. Find critical points: \(x = -1, 1\).
  
4. Determine where the inequality holds:
\[
\text{Solution: } (-\1]


1. (3x - 15) / x <= 5 - x; 2. (4t + 5) / t > 3; 3. (2x) / (x + 1) >= 1

Explore step-by-step solutions to algebraic inequalities involving fractions and quadratics. Learn about critical points, sign analysis, and how to solve inequalities like (3x - 15) / x <= 5 - x, (4t + 5) / t > 3, and (2x) / (x + 1) >= 1. Perfect for high school students in Grades 9-12.

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