Question

Explain and solve (x-7/3x+6 )less than 0

Answer 1

To solve the inequality \(\frac{x - 7}{3x + 6} < 0\), we need to determine when the rational expression is negative. This involves analyzing both the numerator and the denominator.

### Step 1: Identify when the numerator and denominator are zero

**Numerator:**
\[
x - 7 = 0 \implies x = 7
\]

**Denominator:**
\[
3x + 6 = 0 \implies 3x = -6 \implies x = -2
\]

### Step 2: Determine the sign of the expression in different intervals

The critical points we found are \(x = 7\) and \(x = -2\). These points divide the number line into three intervals:
1. \( (-\infty, -2) \)
2. \( (-2, 7) \)
3. \( (7, \infty) \)

Next, we will test a point from each interval to see if the expression is negative in that interval.

**Interval 1: \( (-\infty, -2) \) (e.g., \( x = -3 \))**
\[
\frac{x - 7}{3x + 6} = \frac{-3 - 7}{3(-3) + 6} = \frac{-10}{-9} = \frac{10}{9} > 0
\]

**Interval 2: \( (-2, 7) \) (e.g., \( x = 0 \))**
\[
\frac{x - 7}{3x + 6} = \frac{0 - 7}{3(0) + 6} = \frac{-7}{6} < 0
\]

**Interval 3: \( (7, \infty) \) (e.g., \( x = 8 \))**
\[
\frac{x - 7}{3x + 6} = \frac{8 - 7}{3(8) + 6} = \frac{1}{24} > 0
\]

### Step 3: Compile the results

From our testing:
- The expression is positive in the interval \( (-\infty, -2) \).
- The expression is negative in the interval \( (-2, 7) \).
- The expression is positive in the interval \( (7, \infty) \).

### Step 4: Determine the solution set

Since we want \(\frac{x - 7}{3x + 6} < 0\), we include the interval where the expression is negative and exclude the points where the numerator and denominator equal zero (since the expression is undefined at \(x = -2\)):

Thus, the solution to the inequality is:
\[
\boxed{(-2, 7)}
\]