https://file.aiquickdraw.com/mathbot/file/1728503523877f422bqb2.jpg

The image contains six inequalities to solve. Let's solve them one by one.

### (a) \( 2x + 1 > 5 \)
1. Subtract 1 from both sides:
   \[
   2x > 4
   \]
2. Divide by 2:
   \[
   x > 2
   \]
   
### (b) \( 3x - 8 \leq 13 \)
1. Add 8 to both sides:
   \[
   3x \leq 21
   \]
2. Divide by 3:
   \[
   x \leq 7
   \]

### (c) \( 3 - \frac{2}{x} > x \)
1. Rearrange:
   \[
   3 > x + \frac{2}{x}
   \]
2. Multiply both sides by \( x \) (assuming \( x > 0 \) because of division):
   \[
   3x > x^2 + 2
   \]
   This simplifies to:
   \[
   x^2 - 3x + 2 < 0
   \]
3. Factor the quadratic:
   \[
   (x - 1)(x - 2) < 0
   \]
4. From the sign analysis, the solution is:
   \[
   1 < x < 2
   \]

### (d) \( \frac{1}{2x} \leq \frac{7}{16x} \)
1. Multiply both sides by \( 16x \) (assuming \( x > 0 \)):
   \[
   8 \leq 7
   \]
   This is a contradiction, so **no solution** exists for this inequality.

### (e) \( \frac{x}{2x - 3} \leq 2 \)
1. Multiply both sides by \( 2x - 3 \) (considering the sign of \( 2x - 3 \)):
   \[
   x \leq 2(2x - 3)
   \]
   This simplifies to:
   \[
   x \leq 4x - 6
   \]
2. Subtract \( x \) from both sides:
   \[
   0 \leq 3x - 6
   \]
3. Add 6:
   \[
   6 \leq 3x
   \]
4. Divide by 3:
   \[
   2 \leq x
   \]
5. Now, check the domain. Since \( 2x - 3 > 0 \), \( x > \frac{3}{2} \), so the solution is:
   \[
   x \geq 2
   \]

### (f) \( \frac{x}{3} - \frac{x}{2x} \geq 1 \)
1. Simplify the second term on the left:
   \[
   \frac{x}{2x} = \frac{1}{2}
   \]
   So, the inequality becomes:
   \[
   \frac{x}{3} - \frac{1}{2} \geq 1
   \]
2. Add \( \frac{1}{2} \) to both sides:
   \[
   \frac{x}{3} \geq \frac{3}{2}
   \]
3. Multiply by 3:
   \[
   x \geq \frac{9}{2}
   \]
   So the solution is:
   \[
   x \geq 4.5
   \]

### Summary of Solutions:
(a) \( x > 2 \)  
(b) \( x \leq 7 \)  
(c) \( 1 < x < 2 \)  
(d) No solution  
(e) \( x \geq 2 \)  
(f) \( x \geq 4.5 \)

Would you like further explanations or details on any step?

Here are some related questions you might consider:
1. How do you solve inequalities with quadratic expressions?
2. How do sign changes affect inequality solutions when multiplying or dividing?
3. How do we handle inequalities with rational expressions?
4. Can you explain the interval notation for inequality solutions?
5. What happens when an inequality has no solution?

**Tip:** When solving inequalities with variables in the denominator, always remember to consider restrictions on the domain to avoid division by zero.

Solve the following inequalities:
(a) 2x + 1 > 5
(b) 3x - 8 <= 13
(c) 3 - 2/x > x
(d) 1/(2x) <= 7/(16x)
(e) x/(2x - 3) <= 2
(f) x/3 - x/(2x) >= 1

This problem set includes six inequalities covering algebraic concepts such as linear inequalities, rational expressions, and quadratic inequalities. Detailed step-by-step solutions show how to manipulate and solve these inequalities for Grades 9-12 students.

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