Equations and Inequalities

Solving Inequalities

Inequalities are mathematical statements that compare two values or expressions using symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Solving inequalities involves finding the set of values that satisfy the inequality.

• > (greater than): Indicates that one value is larger than another.

• < (less than): Indicates that one value is smaller than another.

• ≥ (greater than or equal to): Indicates that one value is larger than or equal to another.

• ≤ (less than or equal to): Indicates that one value is smaller than or equal to another.

Interval Notation

Interval notation is a concise way to describe sets of real numbers that satisfy inequalities. It uses parentheses and brackets to indicate whether endpoints are included or excluded.

• (a, b): All real numbers x for which a < x < b (endpoints not included).

• [a, b]: All real numbers x for which a ≤ x ≤ b (endpoints included).

• (a, b]: All real numbers x for which a < x ≤ b.

• [a, b): All real numbers x for which a ≤ x < b.

Solving Linear Inequalities

Linear inequalities are similar to linear equations but use inequality signs. The solution is often expressed in interval notation.

• Example: Solve Step 1: Subtract 2 from both sides: Step 2: Divide both sides by 3: Solution:

• Example: Solve Step 1: Add 5 to both sides: Step 2: Divide both sides by 2: Solution:

Solving Quadratic Inequalities

Quadratic inequalities involve expressions of the form . Solutions are often found by factoring or using the quadratic formula, then testing intervals.

• Example: Solve Step 1: Factor: Step 2: Find zeros: , Step 3: Test intervals:

  • : Both factors negative, product positive

  • : One factor positive, one negative, product negative

  • : Both factors positive, product positive

  Solution:

Solving Equations: Linear, Quadratic, Rational, and Radical

Equations are mathematical statements asserting equality. Different types require different methods:

• Linear Equation: Example: Solution:

• Quadratic Equation: Example: Solution:

• Rational Equation: Contains fractions with variables in the denominator. Example: Solution:

• Radical Equation: Contains square roots or other roots. Example: Solution:

Factoring Techniques

Factoring is a method used to solve equations and simplify expressions. Common techniques include:

• Difference of Squares:

• Sum and Difference of Cubes:

• Grouping: Used for polynomials with four or more terms.

Grouping Example

• Example: Step 1: Factor: Step 2: Set each factor to zero: or Solution: or

Summary Table: Types of Equations and Methods

Additional info:

• Interval notation is essential for expressing solutions to inequalities.

• Factoring is a foundational skill for solving quadratic and higher-order equations.

• Graphical representations, such as the parabola in image_1, help visualize solutions to quadratic inequalities.