Strategies for solving questions related to inequalities and modulus in the mathematics section of the JEE Advanced exam

Introduction:
Inequalities and modulus are important topics in the mathematics section of the JEE Advanced exam. These questions test the candidate's understanding of the properties and solutions of inequalities and modulus equations. To solve these questions effectively, it is necessary to have a clear understanding of the concepts and apply the appropriate strategies. Here are some strategies to solve questions related to inequalities and modulus in the JEE Advanced exam.

1. Understand the concepts:
- Review the basic concepts of inequalities and modulus equations, including the properties and rules.
- Familiarize yourself with the different types of inequalities, such as linear, quadratic, and rational inequalities.
- Understand the concept of modulus and its properties, such as the definition of modulus, modulus inequalities, and modulus equations.

2. Simplify the expressions:
- Simplify the given expressions involving inequalities and modulus to make them easier to work with.
- Use algebraic techniques like factoring, expanding, and canceling out common factors to simplify the expressions.
- Remember to apply the rules of inequalities and modulus when simplifying the expressions.

3. Solve the inequalities:
- Identify the type of inequality (linear, quadratic, rational, etc.) and choose the appropriate method to solve it.
- For linear inequalities, use techniques like graphing, interval notation, or sign analysis.
- For quadratic inequalities, factorize or use the quadratic formula to find the solutions.
- For rational inequalities, find the critical values and use test points to determine the solution intervals.

4. Solve the modulus equations:
- Divide the modulus equation into two cases: one with a positive modulus and another with a negative modulus.
- Solve each case separately by equating the expression inside the modulus to the positive or negative value of the modulus.
- Solve the resulting equations to find the solutions.

5. Apply logic and reasoning:
- Use logical reasoning to eliminate extraneous solutions by checking if they satisfy the original inequality or modulus equation.
- Be careful with the signs when multiplying or dividing by negative values as it can change the direction of the inequality.
- Look for patterns or special cases that can simplify the problem or provide shortcuts to the solution.

Conclusion:
Solving questions related to inequalities and modulus in the mathematics section of the JEE Advanced exam requires a strong understanding of the concepts and the application of appropriate strategies. By following these strategies, candidates can approach these questions systematically and increase their chances of obtaining the correct solutions. Practice is also crucial to improve problem-solving skills in this area.