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Let's solve the two inequalities step by step.

### Inequality 1: \( 2x + 3 > 5x - 12 \)

**Step 1: Isolate the variable \(x\).**

- Subtract \(2x\) from both sides:
  \[
  3 > 3x - 12
  \]
  
- Add 12 to both sides:
  \[
  15 > 3x
  \]

- Divide by 3:
  \[
  5 > x \quad \text{or} \quad x < 5
  \]

**Solution for inequality 1:**
\[
x < 5
\]

---

### Inequality 2: \( -\frac{1}{2}x + 4 \leq 0 \)

**Step 1: Isolate the variable \(x\).**

- Subtract 4 from both sides:
  \[
  -\frac{1}{2}x \leq -4
  \]

- Multiply both sides by \(-2\) (and remember to flip the inequality sign when multiplying by a negative):
  \[
  x \geq 8
  \]

**Solution for inequality 2:**
\[
x \geq 8
\]

---

### Final solutions:
1. \( x < 5 \)
2. \( x \geq 8 \)

Would you like more details on any step or have any questions?

---

Here are five related questions to consider:
1. How do you solve compound inequalities involving "and" or "or"?
2. What changes when multiplying or dividing inequalities by a negative number?
3. How would the solutions change if the inequalities involved fractions with \(x\)?
4. How can you represent the solutions of these inequalities on a number line?
5. What is the process for solving absolute value inequalities?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Solve the following inequalities:
1. 2x + 3 > 5x - 12
2. -1/2 x + 4 <= 0

Learn how to solve linear inequalities like 2x + 3 > 5x - 12 and -1/2x + 4 <= 0 with detailed step-by-step solutions. This tutorial covers the fundamental algebraic steps for isolating variables and the properties of inequalities. Suitable for students in Grades 8-10.

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