Mastering Algebra: Finding Roots and Solutions for CAT 2024

In preparation for the CAT 2024 exam, mastering algebra is crucial, especially when it comes to solving problems involving functions and their roots. This post will guide you through a classical problem involving a modulus function, detailing the steps to find the number of solutions and roots for a given equation.

Problem Statement

The question we will tackle is: How many values of x exist for y = 2? This problem requires us to analyze the function and determine the critical points and their implications on the graph.

Understanding the Function

The function we are dealing with is defined as follows:

[ y = |x| + |x + 2| - 2|x| ]

To solve this, we first need to find the critical points of the function. Critical points are where the function changes behavior, which is essential for understanding the graph.

Finding Critical Points

To find the critical points, we set the expressions inside the modulus to zero:

• For |x|, we have x = 0.
• For |x + 2|, we have x = -2.

Thus, the critical points we have are:

• x = 0
• x = -2

Next, we need to analyze the function further to determine how many critical points exist. We can rewrite the function as:

[ y = |x| + |x + 2| - 2|x| ]

This simplifies to: [ y = -|x| + |x + 2| ]

Analyzing the Function

To analyze the function, we can break it down into intervals based on the critical points:

1. For x < -2
2. For -2 ≤ x < 0
3. For x ≥ 0

In each interval, we will evaluate the function to find the corresponding y-values.

Plotting the Graph

Once we have the critical points, we can plot the graph. The critical points are:

• x = 0 (y = 0)
• x = -2 (y = 0)

Evaluating y-values at Critical Points

1. At x = 0: [ y = |0| + |0 + 2| - 2|0| = 0 + 2 - 0 = 2 ]

2. At x = 2: [ y = |2| + |2 + 2| - 2|2| = 2 + 4 - 4 = 2 ]

3. At x = -2: [ y = |-2| + |-2 + 2| - 2|-2| = 2 + 0 - 4 = -2 ]

4. At x = -2/3: [ y = 2(-2/3) + |-2/3 + 2| - 2|-2/3| = -4/3 + 4/3 - 4/3 = 0 ]

Summary of y-values

From our evaluations, we find:

• At x = 0, y = 2
• At x = 2, y = 2
• At x = -2, y = -2
• At x = -2/3, y = 0

Conclusion: Finding the Number of Solutions

Now that we have plotted the critical points and evaluated the function, we can determine how many solutions exist for y = 2. The graph will intersect the line y = 2 at multiple points, indicating the number of solutions.

Final Analysis

• If y = 2, the graph intersects at two points, indicating two solutions.
• If y is set to a value greater than 8/3, we will find two solutions.
• If y is between 0 and 8/3, we will find four solutions.

This analysis shows that understanding the behavior of the function through critical points and graphing is essential for solving algebraic problems effectively.

Closing Thoughts

Mastering these concepts will not only help you in the CAT 2024 exam but also enhance your overall problem-solving skills in algebra. Keep practicing, and you will find these types of questions easier to tackle.

Stay tuned for more insights and tips on mastering algebra and other subjects for your upcoming exams!