If ` 4 (2 x + 3) gt 5 - x and 5 x - 3 ( 2 x - 7 ) gt 3 x - 1 ` , then x can take which of the following values ?

Options

A. 6
B. `-1`
C. 5
D. `-6`

Answer

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

### Step 1: Solve the first inequality
We start with the first inequality:
\[ 4(2x + 3) > 5 - x \]

1. Distribute the 4:
\[ 8x + 12 > 5 - x \]

2. Add \( x \) to both sides:
\[ 8x + x + 12 > 5 \]
\[ 9x + 12 > 5 \]

3. Subtract 12 from both sides:
\[ 9x > 5 - 12 \]
\[ 9x > -7 \]

4. Divide both sides by 9:
\[ x > -\frac{7}{9} \]
\[ x > -0.777... \] (approximately -0.7)

### Step 2: Solve the second inequality
Now we move on to the second inequality:
\[ 5x - 3(2x - 7) > 3x - 1 \]

1. Distribute the -3:
\[ 5x - 6x + 21 > 3x - 1 \]

2. Combine like terms on the left side:
\[ -x + 21 > 3x - 1 \]

3. Add \( x \) to both sides:
\[ 21 > 4x - 1 \]

4. Add 1 to both sides:
\[ 22 > 4x \]

5. Divide both sides by 4:
\[ \frac{22}{4} > x \]
\[ 5.5 > x \]
\[ x < 5.5 \]

### Step 3: Combine the results
From the two inequalities we have:
1. \( x > -\frac{7}{9} \) (approximately -0.7)
2. \( x < 5.5 \)

### Step 4: Determine the range of x
The solution can be summarized as:
\[ -\frac{7}{9} < x < 5.5 \]

### Step 5: Identify possible values of x
Now we need to check which values of \( x \) satisfy this range. The values must be greater than approximately -0.7 and less than 5.5.