Understanding Expressions And System Of Linear Inequalities

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Unit: Expressions and System of Equations

Understanding Expressions and System of Linear Inequalities

Expressions:

1. Basics: Expressions are mathematical phrases that combine numbers, variables, and operations (like addition, subtraction, multiplication, and division).
2. Types:
   • Algebraic: Contains variables and constants.

                       Example: 3x + 5.

                   Numeric: Purely numerical, without variables.

                Example: 12 – 4.

Types of Expressions:

• Monomial: Single term expression (e.g., 3x, -7).
• Binomial: Two terms expression (e.g., 2x + 5, y – 1).
• Trinomial: Three terms expression (e.g., x² + 3x – 4, a + b + c).
• Polynomial: Any expression with multiple terms (can include monomial, binomial, trinomial).

1. Components:
   • Variables: Represented by letters and can take different values.
   • Constants: Fixed values.
   • Operators: Symbols like +, -, ×, ÷, representing operations.
2. Evaluating Expressions:
   • Substitute the values of variables and simplify using order of operations (PEMDAS/BODMAS).

• PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• BODMAS: Brackets, Orders (exponents), Multiplication and Division (left to right), Addition and Subtraction (left to right)

Systems of Linear Inequalities:

1. Definition: A set of linear inequalities involving the same set of variables.
2. Linear Inequality: An inequality that can be written in the form ax + by ≤ c or ax + by ≥ c.
3. Graphical Representation:
   • Each inequality corresponds to a shaded region on the coordinate plane.
   • The solution is the overlapping/shaded region common to all inequalities.
4. Solution Methods:
   • Graphical Method: Plot each inequality on the coordinate plane and find the overlapping region.
   • Algebraic Method: Solve each inequality separately and then find the common solution region.
5. Types of Solutions:
   • Feasible Region: The set of points that satisfy all the given inequalities.
   • Bounded Region: When the feasible region is finite.
   • Unbounded Region: When the feasible region extends indefinitely.
6. Testing Solutions:
   • Substitute test points into the original inequalities to check if they satisfy all conditions.

Key Tips for Revision:

1. Practice evaluating various types of expressions.
2. Understand the graphical representation of linear inequalities.
3. Practice solving systems of linear inequalities both graphically and algebraically.
4. Test solutions to ensure they satisfy all conditions.