To solve an inequality, the approach depends on the type of inequality and the function involved. Here's a general guide on how to solve different types of inequalities:

### 1. **Linear Inequality**
Example: \( 2x + 3 > 7 \)

**Steps:**
1. **Isolate the variable** on one side:
   \[
   2x > 7 - 3
   \]
   \[
   2x > 4
   \]
2. **Divide by the coefficient** of the variable:
   \[
   x > \frac{4}{2}
   \]
   \[
   x > 2
   \]
   
The solution is \( x > 2 \).

### 2. **Quadratic Inequality**
Example: \( x^2 - 4x - 5 \leq 0 \)

**Steps:**
1. **Factor the quadratic expression** (if possible):
   \[
   (x - 5)(x + 1) \leq 0
   \]
2. **Find the roots** (set each factor equal to zero):
   \[
   x - 5 = 0 \quad \text{or} \quad x + 1 = 0
   \]
   \[
   x = 5 \quad \text{or} \quad x = -1
   \]
3. **Determine the intervals** around the roots:
   - Test points in the intervals \( (-\infty, -1) \), \( (-1, 5) \), and \( (5, \infty) \).
   - The solution to the inequality is where the expression is less than or equal to zero.

**Final Answer**: The solution is \( -1 \leq x \leq 5 \).

### 3. **Absolute Value Inequality**
Example: \( |2x - 3| < 7 \)

**Steps:**
1. **Write the compound inequality**:
   \[
   -7 < 2x - 3 < 7
   \]
2. **Isolate the variable** in the middle:
   \[
   -7 + 3 < 2x < 7 + 3
   \]
   \[
   -4 < 2x < 10
   \]
3. **Divide by the coefficient** of \( x \):
   \[
   -2 < x < 5
   \]

**Final Answer**: The solution is \( -2 < x < 5 \).

### 4. **Rational Inequality**
Example: \( \frac{x + 2}{x - 3} \geq 1 \)

**Steps:**
1. **Rewrite the inequality** to have 0 on one side:
   \[
   \frac{x + 2}{x - 3} - 1 \geq 0
   \]
   \[
   \frac{x + 2 - (x - 3)}{x - 3} \geq 0
   \]
   \[
   \frac{5}{x - 3} \geq 0
   \]
2. **Solve for the critical points**:
   - The inequality is undefined at \( x = 3 \).
   - Analyze the sign of \( \frac{5}{x - 3} \).

**Final Answer**: The solution depends on the sign analysis; usually, this is determined by testing values on intervals divided by critical points.

Would you like a more detailed explanation or a specific type of inequality solved?

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Here are five related questions:

1. How do you solve a system of inequalities?
2. Can you explain the concept of absolute value in more detail?
3. How do you graph a linear inequality on a coordinate plane?
4. What are the steps to solving a quadratic inequality using a sign chart?
5. How do rational inequalities differ from polynomial inequalities?

**Tip:** When solving inequalities, always remember to flip the inequality sign if you multiply or divide by a negative number.

Learn how to solve various types of inequalities including linear, quadratic, absolute value, and rational inequalities with detailed step-by-step solutions. This guide covers essential algebraic concepts and methods for solving each type of inequality. Suitable for high school students and beyond.

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