The inequality x (x + 3) `lt` 10 proved for what value of x ?

Options

A. `x gt 2, x lt – 5`
B. `- 5 lt x lt 2`
C. `- 2 lt x lt 5`
D. `x lt – 2, x gt 5`

Answer

To solve the inequality \( x(x + 3) < 10 \), we will follow these steps:

### Step 1: Rearrange the Inequality
First, we need to rearrange the inequality to one side:
\[ x(x + 3) - 10 < 0 \]
This simplifies to:
\[ x^2 + 3x - 10 < 0 \]

### Step 2: Factor the Quadratic Expression
Next, we will factor the quadratic expression \( x^2 + 3x - 10 \). We need to find two numbers that multiply to \(-10\) and add up to \(3\). The numbers \(5\) and \(-2\) fit this requirement:
\[ (x + 5)(x - 2) < 0 \]

### Step 3: Find the Critical Points
The critical points (where the expression equals zero) are found by setting each factor to zero:
1. \( x + 5 = 0 \) → \( x = -5 \)
2. \( x - 2 = 0 \) → \( x = 2 \)

### Step 4: Test Intervals
Now we will test the intervals defined by the critical points: \( (-\infty, -5) \), \( (-5, 2) \), and \( (2, \infty) \).

- **Interval 1: \( (-\infty, -5) \)**
Choose \( x = -6 \):
\[ (-6 + 5)(-6 - 2) = (-1)(-8) = 8 > 0 \]

- **Interval 2: \( (-5, 2) \)**
Choose \( x = 0 \):
\[ (0 + 5)(0 - 2) = (5)(-2) = -10 < 0 \]

- **Interval 3: \( (2, \infty) \)**
Choose \( x = 3 \):
\[ (3 + 5)(3 - 2) = (8)(1) = 8 > 0 \]

### Step 5: Conclusion
The inequality \( (x + 5)(x - 2) < 0 \) holds true in the interval \( (-5, 2) \). Therefore, the solution for the inequality \( x(x + 3) < 10 \) is:
\[ -5 < x < 2 \]