`|2x-1|=4x+5` has how many numbers in its solution set ?

Options

A. 0
B. 1
C. 2
D. an infinite number

Explanation

To solve the equation \( |2x - 1| = 4x + 5 \) and determine how many numbers are in its solution set, we can follow these steps:

### Step 1: Understand the Absolute Value Equation
The equation \( |2x - 1| = 4x + 5 \) can be split into two cases based on the definition of absolute value.

### Step 2: Set Up the Cases
1. **Case 1:** \( 2x - 1 = 4x + 5 \)
2. **Case 2:** \( 2x - 1 = -(4x + 5) \)

### Step 3: Solve Case 1
For Case 1:
\[ 2x - 1 = 4x + 5 \]
Rearranging gives:
\[ 2x - 4x = 5 + 1 \]
\[ -2x = 6 \]
\[ x = -3 \]

### Step 4: Solve Case 2
For Case 2:
\[ 2x - 1 = -4x - 5 \]
Rearranging gives:
\[ 2x + 4x = -5 + 1 \]
\[ 6x = -4 \]
\[ x = -\frac{2}{3} \]

### Step 5: List the Solutions
The solutions we found are:
1. \( x = -3 \)
2. \( x = -\frac{2}{3} \)

### Step 6: Count the Solutions
We have two distinct solutions: \( x = -3 \) and \( x = -\frac{2}{3} \).

### Conclusion
Thus, the equation \( |2x - 1| = 4x + 5 \) has **2 numbers** in its solution set.

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