Question

solve the rational inequality and graph the solution set on a real number line. express the solution set in interval notation. (\frac{x + 1}{x + 4}<2) 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 (square) (simplify your answer. type your answer in interval notation. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.) b. the solution set is the empty set.

Explanation:

Step1: Subtract 2 from both sides

To solve the inequality \(\frac{x + 1}{x + 4}<2\), we first subtract 2 from both sides to get \(\frac{x + 1}{x + 4}-2<0\).

Step2: Find a common denominator and simplify

The common denominator of \(\frac{x + 1}{x + 4}\) and \(2\) (which is \(\frac{2(x + 4)}{x + 4}\)) is \(x + 4\). So we have \(\frac{x + 1-2(x + 4)}{x + 4}<0\).
Simplify the numerator: \(x + 1-2x - 8=-x - 7\). So the inequality becomes \(\frac{-x - 7}{x + 4}<0\), or \(\frac{-(x + 7)}{x + 4}<0\), which is equivalent to \(\frac{x + 7}{x + 4}>0\) (since multiplying both sides by - 1 reverses the inequality sign).

Step3: Find critical points

The critical points occur where the numerator or the denominator is zero. For the numerator \(x+7 = 0\), we get \(x=-7\). For the denominator \(x + 4=0\), we get \(x=-4\). These critical points divide the real number line into three intervals: \((-\infty,-7)\), \((-7,-4)\), and \((-4,\infty)\).

Step4: Test intervals

• For the interval \((-\infty,-7)\), let's choose \(x=-8\). Then \(\frac{-8 + 7}{-8+4}=\frac{-1}{-4}=\frac{1}{4}>0\), so this interval is part of the solution.
• For the interval \((-7,-4)\), let's choose \(x=-5\). Then \(\frac{-5 + 7}{-5 + 4}=\frac{2}{-1}=-2<0\), so this interval is not part of the solution.
• For the interval \((-4,\infty)\), let's choose \(x = 0\). Then \(\frac{0+7}{0 + 4}=\frac{7}{4}>0\), so this interval is part of the solution.

We also need to check the critical points. At \(x=-7\), the expression \(\frac{x + 7}{x + 4}=0\), and our inequality is \(\frac{x + 7}{x + 4}>0\), so \(x=-7\) is not included. At \(x=-4\), the denominator is zero, so \(x=-4\) is not included.

Answer:

The solution set is \((-\infty,-7)\cup(-4,\infty)\)