Questions: Solve the inequality (x+8)/(x+2) < -2 Give your answer in interval notation.

Transcript text: Solve the inequality $\frac{x+8}{x+2}<-2$ Give your answer in interval notation. $\square$

Solution

Solution Steps

To solve the inequality $\frac{x+8}{x+2} < -2$, we first need to find the critical points by setting the expression equal to $-2$ and solving for $x$. We also need to consider the points where the expression is undefined. Then, we test intervals determined by these critical points to find where the inequality holds true. Finally, we express the solution in interval notation.

Step 1: Identify Critical Points

To solve the inequality $\frac{x+8}{x+2} < -2$, we first find the critical points by setting the expression equal to $-2$:

\[ \frac{x+8}{x+2} = -2 \]

Solving for $x$, we get:

\[ x + 8 = -2(x + 2) \]

\[ x + 8 = -2x - 4 \]

\[ 3x = -12 \]

\[ x = -4 \]

Thus, the critical point is $x = -4$.

Step 2: Identify Points of Discontinuity

The expression $\frac{x+8}{x+2}$ is undefined when the denominator is zero:

\[ x + 2 = 0 \]

\[ x = -2 \]

Thus, the point of discontinuity is $x = -2$.

Step 3: Test Intervals

The critical points and points of discontinuity divide the number line into intervals: $(-\infty, -4)$, $(-4, -2)$, and $(-2, \infty)$. We test each interval to determine where the inequality holds.

For $x \in (-\infty, -4)$, choose $x = -5$: \[ \frac{-5+8}{-5+2} = \frac{3}{-3} = -1 \quad (\text{not less than } -2) \]
For $x \in (-4, -2)$, choose $x = -3$: \[ \frac{-3+8}{-3+2} = \frac{5}{-1} = -5 \quad (\text{less than } -2) \]
For $x \in (-2, \infty)$, choose $x = 0$: \[ \frac{0+8}{0+2} = \frac{8}{2} = 4 \quad (\text{not less than } -2) \]