Problem 63 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of solutions of \(|[x]-2 x|=4\), where \([x]\) is the greatest integer \(\leq x\), is (A) 2 (B) 4 (C) 1 (D) infinite

Step-by-Step Solution

Answer

The number of solutions is 4.

Step 1: Understand the Problem

The given equation is \(|[x]-2x|=4\), where \([x]\) denotes the greatest integer less than or equal to \(x\). We need to find for how many values of \(x\) this equation holds true.

Step 2: Case 1: Solve \([x] - 2x = 4\)

1. Assume \([x] = n\), where \(n\) is an integer.2. The equation becomes \(n - 2x = 4\).3. Solve for \(x\): \[ 2x = n - 4 \] \[ x = \frac{n - 4}{2} \]4. Since \([x] = n\), it implies \(n \leq x < n+1\). Substitute \(x = \frac{n-4}{2}\) and solve: \[ n \leq \frac{n-4}{2} < n+1 \]5. Solve the inequalities to find possible values of \(n\).

Step 3: Solve the Inequalities from Case 1

Let's solve the inequalities separately:1. For \(n \leq \frac{n-4}{2}\), we multiply by 2 to get: \[ 2n \leq n - 4 \] \[ n \leq -4 \] 2. For \(\frac{n-4}{2} < n+1\): \[ n-4 < 2n + 2 \] \[ -4 < n + 2 \] \[ n > -6 \]3. Combine these to find that \(-6 < n \leq -4\), which means \(n = -5\) or \(n = -4\).

Step 4: Verify Solutions for Case 1

For \(n = -5\):- Solve for \(x = \frac{-5-4}{2} = -4.5\).- Check: \([-4.5] = -5\), so \(-5 - 2(-4.5) = -5 + 9 = 4\). Valid solution.For \(n = -4\):- Solve for \(x = \frac{-4-4}{2} = -4\).- Check: \([-4] = -4\), so \(-4 - 2(-4) = -4 + 8 = 4\). Valid solution.

Step 5: Case 2: Solve \([x] - 2x = -4\)

1. Assume \([x] = n\), where \(n\) is an integer.2. The equation becomes \(n - 2x = -4\).3. Solve for \(x\): \[ 2x = n + 4 \] \[ x = \frac{n + 4}{2} \]4. Since \([x] = n\), it implies \(n \leq x < n+1\). Substitute \(x = \frac{n+4}{2}\) and solve: \[ n \leq \frac{n+4}{2} < n+1 \]5. Solve the inequalities to find possible values of \(n\).

Step 6: Solve the Inequalities from Case 2

Let's solve the inequalities separately:1. For \(n \leq \frac{n+4}{2}\), we multiply by 2 to get: \[ 2n \leq n + 4 \] \[ n \leq 4 \] 2. For \(\frac{n+4}{2} < n+1\): \[ n+4 < 2n+2 \] \[ 4 < n+2 \] \[ n > 2 \]3. Combine these to find that \(2 < n \leq 4\), which means \(n = 3\) or \(n = 4\).

Step 7: Verify Solutions for Case 2

For \(n = 3\):- Solve for \(x = \frac{3+4}{2} = 3.5\).- Check: \([3.5] = 3\), so \(3 - 2(3.5) = 3 - 7 = -4\). Valid solution.For \(n = 4\):- Solve for \(x = \frac{4+4}{2} = 4\).- Check: \([4] = 4\), so \(4 - 2(4) = 4 - 8 = -4\). Valid solution.

Step 8: Count the Solutions

From both cases, we have valid solutions for \(x\) at \(-4.5\), \(-4\), \(3.5\), and \(4\). This gives us a total of 4 solutions.

Key Concepts

Absolute Value Equations

Absolute value equations can seem intimidating at first, but with a little practice, you'll find them quite fun to solve! An absolute value equation is one that involves the absolute value function, denoted as \(|x|\), which essentially measures the "distance" of a number from zero on the number line. Thus, \(|x|\) is always non-negative.

When solving absolute value equations, such as \(|a| = b\), where \(b\) is a positive number, we approach it by considering the two possible cases:

• The expression inside the absolute value could directly equal \(b\). So, \(a = b\).
• Alternatively, the expression could be the negative of \(b\). That means \(a = -b\).

Understanding how to set up these cases is key. In our original problem, the expression \(|[x] - 2x| = 4\) was broken down into two cases:

• \([x] - 2x = 4\)
• \([x] - 2x = -4\)

By solving for \(x\) in each case, we can find all the possible solutions.

Solution of Equations

The art of solving equations involves finding all values of the unknowns that make the equation true. It's like solving a puzzle, where each piece needs to fit perfectly to complete the picture.

In the exercise, solving the equation \(|[x] - 2x| = 4\) required careful attention to the structure of the equation by breaking it into cases due to the absolute value. Each case was simplified into inequalities.
For example, for \([x] - 2x = 4\), we had:

• \(2x = n - 4\)
• Then, solving gives \(x = \frac{n - 4}{2}\)

We solved the inequality constraints to find the potential integer values for \([x]\), because the greatest integer function \([x]\) dictates this.

Similarly, in the second case, we set up and solved \([x] - 2x = -4\), leading to another set of solutions within specific ranges. Solving inequalities like these ones helps to systematically find the correct solutions for \(x\) that satisfy all given conditions.

Piecewise Functions

Piecewise functions are a type of function defined by different expressions over different intervals. They can look a bit tricky at first, but their structure actually makes them very adaptable to represent complex situations.

The greatest integer function, \([x]\), is an example of a piecewise function. It provides a great illustration of how piecewise functions work because its definition involves finding the largest integer less than or equal to \(x\). This changes based on which interval \(x\) falls into.

In the context of finding \(|[x] - 2x| = 4\), we dealt with these intervals explicitly:

• Understanding that \([x] = n\), where \(n\) has specific integer values, was essential.
• The function needed different expressions, \([x] - 2x = 4\) and \([x] - 2x = -4\), based on the absolute value constraints.

This approach allows one to evaluate how equations behave under different conditions by pinpointing which piece of the function it belongs to, making solutions much clearer.