If `2 + 4x lt 5 - x//2 and 3x + 3 gt - 5- 3x,` then x can take which of the following values ?

Options

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

Answer

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

### Step 1: Solve the first inequality
The first inequality given is:
\[ 2 + 4x < 5 - \frac{x}{2} \]

**Step 1.1:** Rearrange the inequality to isolate terms involving \(x\) on one side.
- Add \(\frac{x}{2}\) to both sides:
\[ 2 + 4x + \frac{x}{2} < 5 \]

**Step 1.2:** Convert \(4x\) to a fraction with a common denominator:
\[ 4x = \frac{8x}{2} \]
So the inequality becomes:
\[ 2 + \frac{8x}{2} + \frac{x}{2} < 5 \]

**Step 1.3:** Combine like terms:
\[ 2 + \frac{9x}{2} < 5 \]

**Step 1.4:** Subtract 2 from both sides:
\[ \frac{9x}{2} < 3 \]

**Step 1.5:** Multiply both sides by 2 to eliminate the fraction:
\[ 9x < 6 \]

**Step 1.6:** Divide by 9:
\[ x < \frac{6}{9} \]
\[ x < \frac{2}{3} \]
\[ x < 0.66 \]

### Step 2: Solve the second inequality
The second inequality given is:
\[ 3x + 3 > -5 - 3x \]

**Step 2.1:** Rearrange the inequality to isolate terms involving \(x\):
- Add \(3x\) to both sides:
\[ 3x + 3 + 3x > -5 \]

**Step 2.2:** Combine like terms:
\[ 6x + 3 > -5 \]

**Step 2.3:** Subtract 3 from both sides:
\[ 6x > -8 \]

**Step 2.4:** Divide by 6:
\[ x > -\frac{8}{6} \]
\[ x > -\frac{4}{3} \]
\[ x > -1.33 \]

### Step 3: Combine the results
From the two inequalities, we have:
1. \( x < \frac{2}{3} \) (or \( x < 0.66 \))
2. \( x > -\frac{4}{3} \) (or \( x > -1.33 \))

### Step 4: Determine the range of \(x\)
The solution for \(x\) is:
\[ -\frac{4}{3} < x < \frac{2}{3} \]

### Step 5: Identify possible values for \(x\)
Now we need to check which values from the options fall within the range \(-\frac{4}{3} < x < \frac{2}{3}\).