Pre University Remedial Program Mathematics chapter 1 part 1

Introduction

This tutorial provides a structured overview of solving equations and inequalities, focusing on key mathematical concepts such as exponents, radicals, systems of linear equations, and absolute values. It is designed for students in a pre-university remedial mathematics program, offering a clear path to understanding these foundational topics.

Step 1: Understanding Equations Involving Exponents and Radicals

To solve equations that include exponents and radicals, follow these steps:

1. Identify the Equation: Look for terms that involve exponents (e.g., x²) and radicals (e.g., √x).
2. Isolate the Variable:
   • For equations with exponents, apply logarithms if necessary to bring the variable down.
   • For equations with radicals, square both sides to eliminate the radical.
3. Solve the Resulting Equation: After isolating the variable, solve for it using standard algebraic techniques.

Practical Tip

Always check your solutions by substituting them back into the original equation to ensure they satisfy it.

Step 2: Solving Systems of Linear Equations in Two Variables

Systems of linear equations can be solved using various methods. Here’s a common approach:

1. Write the Equations: Ensure both equations are in the format Ax + By = C.
2. Choose a Method: You can use:
   • Substitution: Solve one equation for one variable and substitute into the other.
   • Elimination: Add or subtract equations to eliminate one variable.
3. Solve for Both Variables:
   • Once one variable is found, substitute it back to find the other variable.
4. Check Your Solution: Substitute both values back into the original equations to verify.

Common Pitfall

Be cautious of inconsistent systems that have no solutions or dependent systems that have infinitely many solutions.

Step 3: Working with Equations Involving Absolute Values

Equations with absolute values require special attention. Here’s how to approach them:

1. Set Up Cases: An absolute value equation |x| = a creates two cases:
   • Case 1: x = a
   • Case 2: x = -a
2. Solve Each Case: Solve for x in both scenarios.
3. Check for Extraneous Solutions: After solving, substitute back into the original equation to confirm that each solution is valid.

Practical Advice

Remember that absolute values are always non-negative, so discard any negative solutions that do not apply.

Conclusion

This tutorial covered essential methods for solving equations and inequalities, focusing on exponents, systems of linear equations, and absolute values. Mastering these techniques will provide a solid foundation for further mathematical studies. As a next step, practice by solving various problems related to these topics, ensuring you’re comfortable with each method.