Consider the equation`|2x|-|x-4|=x+4` Total number of prime numbers less than 20 satisfying the equation is

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the equation \( |2x| - |x - 4| = x + 4 \) and find the total number of prime numbers less than 20 that satisfy this equation, we will break down the problem step by step.

### Step 1: Analyze the Absolute Values
We need to consider different cases based on the values of \( x \) because of the absolute value functions.

1. **Case 1:** \( x \geq 4 \)
- Here, both \( |2x| \) and \( |x - 4| \) are positive.
- The equation becomes:
\[ 2x - (x - 4) = x + 4 \]
- Simplifying this:
\[ 2x - x + 4 = x + 4 \]
\[ x + 4 = x + 4 \]
- This is true for all \( x \geq 4 \).

2. **Case 2:** \( 0 \leq x < 4 \)
- Here, \( |2x| \) is positive and \( |x - 4| \) is negative.
- The equation becomes:
\[ 2x - (-(x - 4)) = x + 4 \]
- Simplifying this:
\[ 2x + x - 4 = x + 4 \]
\[ 3x - 4 = x + 4 \]
\[ 3x - x = 4 + 4 \]
\[ 2x = 8 \implies x = 4 \]
- Since \( x = 4 \) is the boundary, we check if it satisfies the original equation:
\[ |2(4)| - |4 - 4| = 4 + 4 \implies 8 - 0 = 8 \]
- This holds true.

3. **Case 3:** \( x < 0 \)
- Here, both \( |2x| \) and \( |x - 4| \) are negative.
- The equation becomes:
\[ -2x - (-(x - 4)) = x + 4 \]
- Simplifying this:
\[ -2x + x - 4 = x + 4 \]
\[ -x - 4 = x + 4 \]
\[ -x - x = 4 + 4 \]
\[ -2x = 8 \implies x = -4 \]
- Check if \( x = -4 \) satisfies the original equation:
\[ |2(-4)| - |-4 - 4| = -4 + 4 \implies 8 - 8 = 0 \]
- This holds true.

### Step 2: Summary of Solutions
From the cases analyzed:
- For \( x \geq 4 \), all \( x \) values satisfy the equation.
- For \( 0 \leq x < 4 \), the only solution is \( x = 4 \).
- For \( x < 0 \), the solution is \( x = -4 \).

Thus, the solutions to the equation are \( x \geq 4 \) and \( x = -4 \).

### Step 3: Find Prime Numbers Less Than 20
Now we need to find the prime numbers less than 20:
- The prime numbers less than 20 are: \( 2, 3, 5, 7, 11, 13, 17, 19 \).

### Step 4: Filter Prime Numbers Greater Than 4
Since we are interested in the prime numbers that satisfy \( x \geq 4 \):
- The relevant prime numbers are: \( 5, 7, 11, 13, 17, 19 \).

### Conclusion
The total number of prime numbers less than 20 that satisfy the equation is **6**.