The least integral value of x for which `33 - x(2 + 3x) gt 0` is

Options

A. 1
B. 2
C. -4
D. -3

Answer

To solve the inequality \( 33 - x(2 + 3x) > 0 \), we will follow these steps:

### Step 1: Rewrite the inequality
Start with the given inequality:
\[ 33 - x(2 + 3x) > 0 \]
Distributing \( x \) gives:
\[ 33 - (2x + 3x^2) > 0 \]
This simplifies to:
\[ 33 - 2x - 3x^2 > 0 \]

### Step 2: Rearranging the inequality
Rearranging the terms, we have:
\[ -3x^2 - 2x + 33 > 0 \]
Multiplying the entire inequality by -1 (and reversing the inequality sign) gives:
\[ 3x^2 + 2x - 33 < 0 \]

### Step 3: Factor the quadratic expression
Next, we need to factor the quadratic \( 3x^2 + 2x - 33 \). To do this, we look for two numbers that multiply to \( 3 \times -33 = -99 \) and add up to \( 2 \). The numbers \( 11 \) and \( -9 \) work:
\[ 3x^2 + 11x - 9x - 33 < 0 \]
Grouping the terms:
\[ (3x^2 + 11x) + (-9x - 33) < 0 \]
Factoring by grouping:
\[ x(3x + 11) - 3(3x + 11) < 0 \]
This can be factored as:
\[ (3x + 11)(x - 3) < 0 \]

### Step 4: Find the critical points
Set each factor to zero to find the critical points:
1. \( 3x + 11 = 0 \) gives \( x = -\frac{11}{3} \)
2. \( x - 3 = 0 \) gives \( x = 3 \)

### Step 5: Determine the intervals
The critical points divide the number line into intervals:
1. \( (-\infty, -\frac{11}{3}) \)
2. \( (-\frac{11}{3}, 3) \)
3. \( (3, \infty) \)

### Step 6: Test the intervals
We will test a point from each interval to see where the inequality holds:
- For \( x = -4 \) (in \( (-\infty, -\frac{11}{3}) \)):
\[ (3(-4) + 11)(-4 - 3) = (-12 + 11)(-7) = (-1)(-7) = 7 > 0 \quad \text{(not valid)} \]
- For \( x = 0 \) (in \( (-\frac{11}{3}, 3) \)):
\[ (3(0) + 11)(0 - 3) = (11)(-3) = -33 < 0 \quad \text{(valid)} \]
- For \( x = 4 \) (in \( (3, \infty) \)):
\[ (3(4) + 11)(4 - 3) = (12 + 11)(1) = 23 > 0 \quad \text{(not valid)} \]

### Step 7: Conclusion
The valid interval for the inequality \( 3x^2 + 2x - 33 < 0 \) is:
\[ -\frac{11}{3} < x < 3 \]
The least integral value of \( x \) in this interval is:
\[ -3 \]

### Final Answer
The least integral value of \( x \) for which \( 33 - x(2 + 3x) > 0 \) is:
\[ \boxed{-3} \]