Inequalities are mathematical expressions that compare the relative sizes or values of two quantities. They are used to represent relationships where one value is greater than, less than, or not equal to another. Inequalities are essential in various fields of mathematics and have practical applications in real-life problem-solving.

1. Types of Inequalities

a. Linear Inequalities: - These are inequalities involving linear expressions (polynomials of degree 1). - General form: ax + b < c or ax + b > c - Example: 2x - 3 < 7

b. Quadratic Inequalities: - These are inequalities involving quadratic expressions (polynomials of degree 2). - General form: ax2 + bx + c < 0 or ax2 + bx + c > 0 - Example: x2 - 4x > 0

c. Rational Inequalities: - These involve rational expressions (fractions). - General form: (ax + b) / (cx + d) < 0 or (ax + b) / (cx + d) > 0 - Example: (2x - 1) / (x + 3) < 0

2. Solving Inequalities

a. Isolating the Variable: - As with equations, the first step is to isolate the variable on one side of the inequality. - Use inverse operations (addition, subtraction, multiplication, division) to do this.

b. Balancing Inequalities: - When performing operations on both sides, remember to flip the inequality sign when multiplying or dividing by a negative number.

c. Graphical Representation: - Plot the inequality on a number line or a Cartesian plane to visualize the solution set.

3. Inequality Symbols

a. Less Than (<): - Represents that the left side is smaller than the right side.

b. Greater Than (>): - Represents that the left side is larger than the right side.

c. Less Than or Equal To (≤): - Indicates that the left side is less than or equal to the right side.

d. Greater Than or Equal To (≥): - Shows that the left side is greater than or equal to the right side.

4. Solutions to Inequalities

a. Interval Notation: - Inequalities are often expressed using interval notation to denote solution sets. - Example: (1, 5) represents all numbers between 1 and 5, not including 1 and 5.

b. Set Notation: - Set-builder notation can also be used to describe solution sets. - Example: {x | 1 < x < 5} represents the set of all x such that 1 < x < 5.

5. Compound Inequalities:

  • Compound inequalities involve multiple inequalities combined with logical operators "and" (conjunction) or "or" (disjunction).
  • Example: 2 < x < 5 (and) x < -3 or x > 3 (or)

6. Word Problems:

  • Many real-life problems can be modeled and solved using inequalities. Translate the problem into an inequality, solve it, and interpret the results in the context of the problem.

7. Systems of Inequalities:

  • Systems of inequalities involve multiple inequalities with multiple variables. The solution is a set of values that satisfy all the inequalities simultaneously.