Inequalities are mathematical expressions that involve a comparison between two quantities. Instead of an equation, which asserts that two expressions are equal, an inequality shows that one expression is greater than or less than the other.

The common inequality symbols are:

1. > (Greater than)
2. < (Less than)
3. ≥ (Greater than or equal to)
4. ≤ (Less than or equal to)
5. ≠ (Not equal to)

Types of Inequalities

1. Linear Inequalities: These are inequalities where the variable appears in a linear form (i.e., the variable is to the power of 1). For example:

   • $2x + 3 > 5$

2. Quadratic Inequalities: These involve a quadratic expression on one side of the inequality. For example:

   • $x^2 - 4x + 3 < 0$

3. Compound Inequalities: These are formed by combining two or more inequalities, often using "and" or "or". For example:

   • $1 < x \leq 5$

4. Absolute Value Inequalities: These involve an absolute value, such as:

   • $|x - 3| \geq 5$

Solving Inequalities

The approach to solving inequalities is similar to solving equations, with a few key differences:

1. Isolate the variable: Solve the inequality for the variable, just like you would in an equation. Example: $2x + 3 > 5 \implies 2x > 2 \implies x > 1$

2. Multiply or divide by negative numbers: When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol must flip. Example: $-2x > 4 \implies x < -2 \quad (\text{flip the inequality symbol})$

3. Graphical Representation: Inequalities can be graphed on a number line. For example:

   • $x > 1$ would be represented as an open circle at $x = 1$ and a shaded region to the right of it.
   • $x \geq 1$ would have a closed circle and a shaded region to the right.

Would you like to solve an inequality together or need further explanation on any of these topics?

Related questions:

1. How do I solve quadratic inequalities?
2. What is the difference between strict and non-strict inequalities?
3. How do I graph compound inequalities on a number line?
4. What happens when we solve absolute value inequalities?
5. How do inequalities appear in real-world applications?

Tip: Always pay attention to the direction of the inequality when multiplying or dividing by a negative number!