Questions: Solve the inequality (x+4)/(x+7) > -3 Give your answer in interval notation.

Solution Steps

To solve the inequality \(\frac{x+4}{x+7} > -3\), we first move all terms to one side to form a single rational expression. Then, we find the critical points by setting the numerator and denominator to zero. We test intervals between these critical points to determine where the inequality holds true. Finally, we express the solution in interval notation.

We start with the inequality

\[ \frac{x+4}{x+7} > -3. \]

To solve this, we rearrange it to form a single rational expression:

\[ \frac{x+4}{x+7} + 3 > 0. \]

This simplifies to

\[ \frac{x+4 + 3(x+7)}{x+7} > 0, \]

which further simplifies to

\[ \frac{4x + 25}{x + 7} > 0. \]

Next, we find the critical points by setting the numerator and denominator to zero.

1. The numerator \(4x + 25 = 0\) gives us

\[ x = -\frac{25}{4}. \]

2. The denominator \(x + 7 = 0\) gives us

\[ x = -7. \]

Thus, the critical points are \(x = -\frac{25}{4}\) and \(x = -7\).

We now test the intervals determined by the critical points \((- \infty, -7)\), \((-7, -\frac{25}{4})\), and \((- \frac{25}{4}, \infty)\) to see where the inequality holds.

• For the interval \((- \infty, -7)\), choose \(x = -8\): \[ \frac{4(-8) + 25}{-8 + 7} = \frac{-32 + 25}{-1} = \frac{-7}{-1} = 7 > 0. \] This interval satisfies the inequality.

• For the interval \((-7, -\frac{25}{4})\), choose \(x = -6\): \[ \frac{4(-6) + 25}{-6 + 7} = \frac{-24 + 25}{1} = 1 > 0. \] This interval also satisfies the inequality.

• For the interval \((- \frac{25}{4}, \infty)\), choose \(x = 0\): \[ \frac{4(0) + 25}{0 + 7} = \frac{25}{7} > 0. \] This interval satisfies the inequality as well.

The solution to the inequality is the union of the intervals where the inequality holds true. However, we must exclude the point where the denominator is zero, which is \(x = -7\). Therefore, the solution in interval notation is:

\[ (-\infty, -7) \cup (-7, -\frac{25}{4}) \cup (-\frac{25}{4}, \infty). \]

Final Answer