To solve the rational inequality
[tex]\[ \frac{x+4}{x+9} < 2, \][/tex]
we need to follow a logical step-by-step approach:
1. Rewrite the Inequality:
First, we start by rewriting the inequality in the form:
[tex]\[ \frac{x+4}{x+9} - 2 < 0. \][/tex]
2. Common Denominator:
To combine the terms, we need a common denominator. The expression becomes:
[tex]\[ \frac{x+4 - 2(x+9)}{x+9} < 0. \][/tex]
3. Simplify the Numerator:
Simplify the numerator by distributing and combining like terms:
[tex]\[ \frac{x + 4 - 2x - 18}{x+9} < 0 \][/tex]
[tex]\[ \frac{-x - 14}{x+9} < 0. \][/tex]
4. Critical Points:
The critical points are found where the numerator and denominator equal zero:
[tex]\[ -x - 14 = 0 \quad \Rightarrow \quad x = -14, \][/tex]
[tex]\[ x + 9 = 0 \quad \Rightarrow \quad x = -9. \][/tex]
5. Sign Analysis:
Next, we determine the sign of the expression [tex]\(\frac{-x - 14}{x+9}\)[/tex] on the intervals defined by the critical points [tex]\(x = -14\)[/tex] and [tex]\(x = -9\)[/tex]:
- For [tex]\(x < -14\)[/tex] (e.g., [tex]\(x = -15\)[/tex]):
[tex]\[ \frac{-(-15) - 14}{-15 + 9} = \frac{15 - 14}{-6} = \frac{1}{-6} = -\frac{1}{6} < 0. \][/tex]
- For [tex]\(-14 < x < -9\)[/tex] (e.g., [tex]\(x = -10\)[/tex]):
[tex]\[ \frac{-(-10) - 14}{-10 + 9} = \frac{10 - 14}{-1} = \frac{-4}{-1} = 4 > 0. \][/tex]
- For [tex]\(x > -9\)[/tex] (e.g., [tex]\(x = 0\)[/tex]):
[tex]\[ \frac{-0 - 14}{0 + 9} = \frac{-14}{9} = -\frac{14}{9} < 0. \][/tex]
6. Combine the Intervals:
From the sign analysis, we determine where the expression is negative:
- The interval [tex]\((- \infty, -14)\)[/tex],
- The interval [tex]\((-9, \infty)\)[/tex].
Therefore, the solution set in interval notation where the given inequality holds true is:
[tex]\[ (-\infty, -14) \cup (-9, \infty). \][/tex]
The graph of the solution set on a real number line would have open intervals (not including the points [tex]\(-14\)[/tex] and [tex]\(-9\)[/tex]) indicating the regions where the inequality is satisfied.
Thus, the correct choice is:
A. The solution set is [tex]\((- \infty, -14) \cup (-9, \infty)\)[/tex].
Graphically, this can be represented as:
```
<---(-∞)---(-14)---(-9)---(∞)--->
```
The points [tex]\(-14\)[/tex] and [tex]\(-9\)[/tex] are not included in the solution set as indicated by the open intervals.
