If `x + 2 (3-5x) gt -1 - 2x gt 5 - x//3,` then the value of x is (a) -3 (b) -4 (c) 2 (d) 3

Options

A. `-3`
B. `-4`
C. 2
D. 3

Answer

To solve the inequality \( x + 2(3 - 5x) > -1 - 2x > 5 - \frac{x}{3} \), we will break it down into two parts and solve each inequality step by step.

### Step 1: Solve the first part of the inequality
We start with the first part of the inequality:
\[ x + 2(3 - 5x) > -1 - 2x \]

1. Distribute \( 2 \) in the left side:
\[ x + 6 - 10x > -1 - 2x \]

2. Combine like terms:
\[ -9x + 6 > -1 - 2x \]

3. Add \( 9x \) to both sides:
\[ 6 > 7x - 1 \]

4. Add \( 1 \) to both sides:
\[ 7 > 7x \]

5. Divide by \( 7 \):
\[ 1 > x \]
or
\[ x < 1 \]

### Step 2: Solve the second part of the inequality
Now we solve the second part:
\[ -1 - 2x > 5 - \frac{x}{3} \]

1. Add \( 2x \) to both sides:
\[ -1 > 5 + 2x - \frac{x}{3} \]

2. Combine the terms involving \( x \):
- Convert \( 2x \) to a fraction:
\[ 2x = \frac{6x}{3} \]
- Now combine:
\[ -1 > 5 + \frac{6x - x}{3} \]
\[ -1 > 5 + \frac{5x}{3} \]

3. Subtract \( 5 \) from both sides:
\[ -6 > \frac{5x}{3} \]

4. Multiply both sides by \( 3 \):
\[ -18 > 5x \]

5. Divide by \( 5 \):
\[ -\frac{18}{5} > x \]
or
\[ x < -\frac{18}{5} \]

### Step 3: Combine the results
From the two inequalities, we have:
1. \( x < 1 \)
2. \( x < -\frac{18}{5} \) (which is approximately \( -3.6 \))

Since \( -\frac{18}{5} \) is less than \( 1 \), the more restrictive condition is \( x < -\frac{18}{5} \).

### Conclusion
The possible values of \( x \) must be less than \( -\frac{18}{5} \). Among the options given:
- (a) -3
- (b) -4
- (c) 2
- (d) 3

The only option that satisfies \( x < -3.6 \) is:
**(b) -4**