For a non-zero vector a, the set of real number, satisfying `|(5-x)a|lt|2a|` consists of all x such that

Options

A. `0 lt x lt 3`
B. `3 lt x lt 7`
C. `-7 lt x lt -3`
D. `-7 lt x lt 3`

Explanation

To solve the inequality \( |(5-x)\mathbf{a}| < |2\mathbf{a}| \), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ |(5-x)\mathbf{a}| < |2\mathbf{a}| \]
Since \(\mathbf{a}\) is a non-zero vector, we can simplify this using the property of magnitudes:
\[ |(5-x)| \cdot |\mathbf{a}| < |2| \cdot |\mathbf{a}| \]
This simplifies to:
\[ |(5-x)| < 2 \]

### Step 2: Remove the absolute value
The inequality \( |(5-x)| < 2 \) implies two inequalities:
\[ -2 < 5 - x < 2 \]

### Step 3: Solve the left inequality
Starting with the left part of the compound inequality:
\[ -2 < 5 - x \]
Rearranging gives:
\[ x < 5 + 2 \implies x < 7 \]

### Step 4: Solve the right inequality
Now, we solve the right part of the compound inequality:
\[ 5 - x < 2 \]
Rearranging gives:
\[ -x < 2 - 5 \implies -x < -3 \implies x > 3 \]

### Step 5: Combine the results
Combining the results from steps 3 and 4, we have:
\[ 3 < x < 7 \]

### Final Answer
The set of real numbers satisfying the original inequality is:
\[ x \in (3, 7) \]