If `5 - x lt 1 + 5x and 2x + 3 (5 - 2x) lt 2 - 3x,` then x can ake which of the following values ?

Options

A. 0
B. `-1`
C. 14
D. 1

Answer

To solve the inequalities given in the question, we will break it down into two parts.

### Step 1: Solve the first inequality
The first inequality is:
\[ 5 - x < 1 + 5x \]

1. Start by isolating \( x \). First, subtract \( 1 \) from both sides:
\[ 5 - x - 1 < 5x \]
\[ 4 - x < 5x \]

2. Next, add \( x \) to both sides:
\[ 4 < 5x + x \]
\[ 4 < 6x \]

3. Now, divide both sides by \( 6 \):
\[ \frac{4}{6} < x \]
\[ \frac{2}{3} < x \]

This means:
\[ x > \frac{2}{3} \]

### Step 2: Solve the second inequality
The second inequality is:
\[ 2x + 3(5 - 2x) < 2 - 3x \]

1. Start by distributing \( 3 \) in the left side:
\[ 2x + 15 - 6x < 2 - 3x \]

2. Combine like terms on the left side:
\[ 15 - 4x < 2 - 3x \]

3. Now, add \( 4x \) to both sides:
\[ 15 < 2 + x \]

4. Subtract \( 2 \) from both sides:
\[ 15 - 2 < x \]
\[ 13 < x \]

This means:
\[ x > 13 \]

### Step 3: Combine the results
From the two inequalities we have:
1. \( x > \frac{2}{3} \)
2. \( x > 13 \)

Since \( 13 \) is greater than \( \frac{2}{3} \), the more restrictive condition is \( x > 13 \).

### Conclusion
Thus, the values that \( x \) can take must be greater than \( 13 \). The only integer value that satisfies this condition from the options provided is \( 14 \).

### Final Answer
The value of \( x \) that satisfies both inequalities is \( 14 \).

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