Solving Linear Equations and Linear Inequalities: Basic Example
Key Concepts
Linear Equations and Inequalities: Focus on solving equations and inequalities of the form "3l - 6 ≥ 8".
Objective: Isolate the variable (l) on one side of the equation or inequality.
Steps for Solving
Eliminate the Constant:
Original Inequality: 3l - 6 ≥ 8.
Add 6 to both sides to eliminate "-6":
Resulting Inequality: 3l ≥ 14.
Isolate the Variable:
Divide both sides by 3 to solve for l:
Resulting Inequality: l ≥ 14/3.
Order of Operations:
Use the reverse order of operations, known as SADMEP (Subtraction, Addition, Division, Multiplication, Exponents, Parentheses), instead of the standard PEMDAS.
Properties of Inequalities:
If you multiply or divide both sides by a negative number, flip the inequality sign.
In this case, since division was by a positive number, the inequality direction remains unchanged.
Important Points
Consistency in Equations: When performing operations, ensure changes are made to both sides to maintain equality/inequality.
Maintaining Inequality: Understanding when and why to flip the inequality sign is crucial, particularly when dealing with negative coefficients.
Common Queries and Clarifications
Why add or subtract from both sides?
To maintain the balance of the equation/inequality.
When to flip the inequality sign?
Only when multiplying or dividing by a negative.
Using SADMEP: Reverse order of operations when solving inequalities to isolate the variable effectively.
Student FAQs and Answers
Why can't you just add to one side?
Adding to one side disrupts the balance of the equation, leading to incorrect solutions.
Does "greater than or equal to" affect solving the equation?
It is treated similarly to an equals sign during the algebraic manipulation.
Conclusion
Solving linear equations and inequalities involves systematic steps to isolate the variable while respecting operational rules and properties. Understanding these principles is essential for accurate problem-solving in algebra.