QuestionQuestion 7 (1 point) Find the solution set for $\frac{x^{2}}{x+3}<\frac{9}{x+3}$ .

Studdy Solution

STEP 1

1. The inequality given is $\frac{x^2}{x+3} < \frac{9}{x+3}$ .
2. We assume $x \neq -3$ to avoid division by zero.
3. We will solve the inequality by considering the critical points and testing intervals.

STEP 2

1. Simplify the inequality.
2. Identify critical points.
3. Test intervals to determine where the inequality holds.
4. Write the solution set.

STEP 3

First, simplify the inequality by multiplying both sides by $x + 3$ , assuming $x \neq -3$ :
$\frac{x^2}{x+3} < \frac{9}{x+3}$
Multiply both sides by $x + 3$ :
$x^2 < 9$

STEP 4

Identify the critical points by solving the equation $x^2 = 9$ :
$x^2 = 9$
Taking the square root of both sides gives:
$x = 3 \quad \text{or} \quad x = -3$

STEP 5

Consider the intervals determined by the critical points: $(- \infty, -3)$ , $(-3, 3)$ , and $(3, \infty)$ .
- For $x \in (-\infty, -3)$ : Choose a test point, say $x = -4$ . $ (-4)^2 = 16 \quad \text{and} \quad 16 < 9 \quad \text{is false.} \]
- For $x \in (-3, 3)$ : Choose a test point, say $x = 0$ . $ 0^2 = 0 \quad \text{and} \quad 0 < 9 \quad \text{is true.} \]
- For $x \in (3, \infty)$ : Choose a test point, say $x = 4$ . $ 4^2 = 16 \quad \text{and} \quad 16 < 9 \quad \text{is false.} \]

STEP 6

Since the inequality holds only in the interval $(-3, 3)$ , and $x \neq -3$ due to division by zero, the solution set is:
$(-3, 3)$
The solution set for the inequality is:
$\boxed{(-3, 3)}$

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