Solving Compound Inequalities in Class 11 Mathematics

In the realm of mathematics, understanding inequalities is crucial for building a strong foundation in algebra. This article will delve into the sequence of steps needed to solve the compound inequality featured in NCERT Class 11 Mathematics Chapter 6, Exercise 6.4. Specifically, we will focus on question 1 of this exercise which is: 2 ≤ 3x–4 ≤ 5.

Understanding Compound Inequalities

A compound inequality is a combination of two or more inequalities joined by the words 'and' or 'or'. In the problem at hand, we deal with a compound inequality that connects two separate inequalities with 'and': (2 leq 3x - 4 leq 5).

Step-by-Step Solution

Step 1: Add 4 on both sides of the inequality.

Starting with the given inequality, 2 ≤ 3x - 4 ≤ 5, we need to isolate the term with the variable (3x) on the right side. To achieve this, we add 4 to both sides of the inequality. This is a common algebraic operation that maintains the inequality's balance.

2 4 ≤ 3x - 4 4 ≤ 5 4

Simplifying the left and right sides: 6 ≤ 3x ≤ 9

Step 2: Divide by 3 on both sides.

Now that we have 6 ≤ 3x ≤ 9, the next step is to divide every term in the inequality by 3. This will isolate the variable (x) on the middle term of the inequality. Once again, this operation is performed on both sides of the inequality to maintain its balance.

6/3 ≤ 3x/3 ≤ 9/3

Simplifying the terms, we get: 2 ≤ x ≤ 3

Step 3: Express the Solution in Interval Form.

The solution 2 ≤ x ≤ 3 can be expressed in interval notation, which is a concise way to represent solutions to inequalities. In this case, the interval is [2, 3]. The square brackets indicate that the endpoints (2 and 3) are included in the interval.

x ∈ [2, 3]

Explanation of the Result

The outcome of the inequality shows that (x) is greater than or equal to 2 but less than or equal to 3. This means that (x) can take any value within the closed interval from 2 to 3, including the endpoints themselves.

Conclusion

Understanding and solving compound inequalities like the one presented in NCERT Class 11 Mathematics Exercise 6.4, Question 1, not only enhances algebraic skills but also sets the stage for more advanced mathematics concepts. Practice with such problems will strengthen your ability to handle complex algebraic expressions and inequalities.

Additional Resources

For further assistance, check out the following resources:

Numerous online tutorials and video explanations. Interactive problem-solving platforms for additional practice. Your textbook and teacher for more examples and exercises.