If `13 x - 1 lt 5x + 2 and 3x + 5 lt 6 - 4x,` then x can take which of the following values ? (a) -1 (b) 1 (c) 2 (d) -2

Options

A. `-1`
B. 1
C. 2
D. `-2`

Answer

To solve the inequalities given in the question, we will break it down step by step.

### Step 1: Solve the first inequality
The first inequality is:
\[ 13x - 1 < 5x + 2 \]

**Rearranging the inequality:**
1. Subtract \(5x\) from both sides:
\[ 13x - 5x - 1 < 2 \]
\[ 8x - 1 < 2 \]

2. Add \(1\) to both sides:
\[ 8x < 3 \]

3. Divide both sides by \(8\):
\[ x < \frac{3}{8} \]

### Step 2: Solve the second inequality
The second inequality is:
\[ 3x + 5 < 6 - 4x \]

**Rearranging the inequality:**
1. Add \(4x\) to both sides:
\[ 3x + 4x + 5 < 6 \]
\[ 7x + 5 < 6 \]

2. Subtract \(5\) from both sides:
\[ 7x < 1 \]

3. Divide both sides by \(7\):
\[ x < \frac{1}{7} \]

### Step 3: Combine the results
Now we have two conditions:
1. \( x < \frac{3}{8} \)
2. \( x < \frac{1}{7} \)

Since \(\frac{1}{7} \approx 0.142857\) and \(\frac{3}{8} = 0.375\), the more restrictive condition is:
\[ x < \frac{1}{7} \]

### Step 4: Determine which values satisfy the condition
Now we check which of the given options satisfy \( x < \frac{1}{7} \):
- (a) -1: Yes, -1 < \(\frac{1}{7}\)
- (b) 1: No, 1 > \(\frac{1}{7}\)
- (c) 2: No, 2 > \(\frac{1}{7}\)
- (d) -2: Yes, -2 < \(\frac{1}{7}\)

### Conclusion
The values that satisfy the inequalities are:
- (a) -1
- (d) -2