Question
Select the number line that represents the solution of x
for each inequality. The red line shows the solution set.
x2<3
(1 point)
Responses
for each inequality. The red line shows the solution set.
x2<3
(1 point)
Responses
Answers
GPT-4o mini
To solve the inequality \( x^2 < 3 \), we start by rewriting it as:
\[
x^2 - 3 < 0
\]
Next, we find the points where \( x^2 - 3 = 0 \) by solving:
\[
x^2 = 3
\]
Taking the square root of both sides, we have:
\[
x = \sqrt{3} \quad \text{and} \quad x = -\sqrt{3}
\]
These points, \( -\sqrt{3} \) and \( \sqrt{3} \), divide the number line into three intervals:
1. \( (-\infty, -\sqrt{3}) \)
2. \( (-\sqrt{3}, \sqrt{3}) \)
3. \( (\sqrt{3}, \infty) \)
To determine which intervals satisfy the inequality \( x^2 < 3 \), we can test a value from each interval:
1. For \( x = -2 \) (in the interval \( (-\infty, -\sqrt{3}) \)):
\[
(-2)^2 = 4 \quad (\text{not a solution, since } 4 \nless 3)
\]
2. For \( x = 0 \) (in the interval \( (-\sqrt{3}, \sqrt{3}) \)):
\[
0^2 = 0 \quad (\text{solution, since } 0 < 3)
\]
3. For \( x = 2 \) (in the interval \( (\sqrt{3}, \infty) \)):
\[
2^2 = 4 \quad (\text{not a solution, since } 4 \nless 3)
\]
Thus, the solution to the inequality \( x^2 < 3 \) is:
\[
-\sqrt{3} < x < \sqrt{3}
\]
On a number line, this is represented by a line segment between \( -\sqrt{3} \) and \( \sqrt{3} \), with open endpoints (indicating that \( -\sqrt{3} \) and \( \sqrt{3} \) are not included in the solution).
If you have a specific number line image or representation to refer to with your question, please provide it, and I can help you identify the correct one.
\[
x^2 - 3 < 0
\]
Next, we find the points where \( x^2 - 3 = 0 \) by solving:
\[
x^2 = 3
\]
Taking the square root of both sides, we have:
\[
x = \sqrt{3} \quad \text{and} \quad x = -\sqrt{3}
\]
These points, \( -\sqrt{3} \) and \( \sqrt{3} \), divide the number line into three intervals:
1. \( (-\infty, -\sqrt{3}) \)
2. \( (-\sqrt{3}, \sqrt{3}) \)
3. \( (\sqrt{3}, \infty) \)
To determine which intervals satisfy the inequality \( x^2 < 3 \), we can test a value from each interval:
1. For \( x = -2 \) (in the interval \( (-\infty, -\sqrt{3}) \)):
\[
(-2)^2 = 4 \quad (\text{not a solution, since } 4 \nless 3)
\]
2. For \( x = 0 \) (in the interval \( (-\sqrt{3}, \sqrt{3}) \)):
\[
0^2 = 0 \quad (\text{solution, since } 0 < 3)
\]
3. For \( x = 2 \) (in the interval \( (\sqrt{3}, \infty) \)):
\[
2^2 = 4 \quad (\text{not a solution, since } 4 \nless 3)
\]
Thus, the solution to the inequality \( x^2 < 3 \) is:
\[
-\sqrt{3} < x < \sqrt{3}
\]
On a number line, this is represented by a line segment between \( -\sqrt{3} \) and \( \sqrt{3} \), with open endpoints (indicating that \( -\sqrt{3} \) and \( \sqrt{3} \) are not included in the solution).
If you have a specific number line image or representation to refer to with your question, please provide it, and I can help you identify the correct one.