QuestionSolve the inequality $\frac{x+8}{x+1}<-4$ Give your answer in interval notation.

Studdy Solution

STEP 1

What is this asking? We need to find all the $x$ values that make the fraction $\frac{x+8}{x+1}$ less than negative four. Watch out! Don't multiply both sides by $x+1$ right away!
We don't know if $x+1$ is positive or negative, so we don't know whether to flip the inequality sign.

STEP 2

1. Rewrite the inequality
2. Find the critical points
3. Test the intervals

STEP 3

Let's move everything to one side to get a zero on the other.
We'll add four to both sides of our inequality: $\frac{x+8}{x+1} + 4 < 0$

STEP 4

Now, let's combine the terms on the left side into a single fraction.
Remember, $4$ is the same as $\frac{4(x+1)}{x+1}$ .
So, we have: $\frac{x+8}{x+1} + \frac{4(x+1)}{x+1} < 0$ $\frac{x+8 + 4(x+1)}{x+1} < 0$ $\frac{x+8 + 4x+4}{x+1} < 0$ $\frac{5x+12}{x+1} < 0$ Much better!

STEP 5

Critical points are where the numerator is zero or the denominator is zero.
The numerator, $5x+12$ , is zero when $5x = -12$ , which means $x = -\frac{12}{5}$ .
The denominator, $x+1$ , is zero when $x = -1$ .
So, our critical points are $x = -\frac{12}{5}$ and $x = -1$ .

STEP 6

Our critical points divide the number line into three intervals: $(-\infty, -\frac{12}{5})$ , $(-\frac{12}{5}, -1)$ , and $(-1, \infty)$ .
Let's test a point in each interval to see if the inequality holds true.

STEP 7

* Interval 1: $(-\infty, -\frac{12}{5})$ .
Let's pick $x = -3$ .
Plugging this into $\frac{5x+12}{x+1}$ , we get $\frac{5(-3)+12}{-3+1} = \frac{-3}{-2} = \frac{3}{2}$ , which is not less than zero.
So, this interval doesn't work.

STEP 8

* Interval 2: $(-\frac{12}{5}, -1)$ .
Let's pick $x = -2$ .
Plugging this in, we get $\frac{5(-2)+12}{-2+1} = \frac{2}{-1} = -2$ , which is less than zero.
This interval works!

STEP 9

* Interval 3: $(-1, \infty)$ .
Let's pick $x = 0$ .
Plugging this in, we get $\frac{5(0)+12}{0+1} = \frac{12}{1} = 12$ , which is not less than zero.
So, this interval doesn't work.

STEP 10

The solution to the inequality is the interval $(-\frac{12}{5}, -1)$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms