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Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation.

[tex]\[ \frac{x+4}{x+9} \ \textless \ 2 \][/tex]

Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The solution set is [tex]$\square$[/tex].
(Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any number.)

B. The solution set is the empty set.

Sagot :

GinnyAnsweridn
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.
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