Question
the graph shows the function y=3x-1 what are the coordinates of the interepts?
Answers
Answer
open or closed circle? and what way left or right for n<-3?
Answer
what is the value of the expression? 2[32-(4-1)^3]
Answered by
GPT-4o mini
To find the intercepts of the function \( y = 3x - 1 \), we need to determine both the x-intercept and the y-intercept.
### 1. Y-Intercept:
The y-intercept is found by setting \( x = 0 \):
\[
y = 3(0) - 1 = -1
\]
Thus, the y-intercept is at the point \( (0, -1) \).
### 2. X-Intercept:
The x-intercept is found by setting \( y = 0 \):
\[
0 = 3x - 1
\]
Solving for \( x \):
\[
3x = 1 \\
x = \frac{1}{3}
\]
Thus, the x-intercept is at the point \( \left( \frac{1}{3}, 0 \right) \).
### Summary of Intercepts:
- Y-Intercept: \( (0, -1) \)
- X-Intercept: \( \left( \frac{1}{3}, 0 \right) \)
### 1. Y-Intercept:
The y-intercept is found by setting \( x = 0 \):
\[
y = 3(0) - 1 = -1
\]
Thus, the y-intercept is at the point \( (0, -1) \).
### 2. X-Intercept:
The x-intercept is found by setting \( y = 0 \):
\[
0 = 3x - 1
\]
Solving for \( x \):
\[
3x = 1 \\
x = \frac{1}{3}
\]
Thus, the x-intercept is at the point \( \left( \frac{1}{3}, 0 \right) \).
### Summary of Intercepts:
- Y-Intercept: \( (0, -1) \)
- X-Intercept: \( \left( \frac{1}{3}, 0 \right) \)
Answered by
GPT-4o mini
The notation \( n < -3 \) indicates an inequality where the values of \( n \) are less than \( -3 \). When graphing this inequality on a number line or a coordinate plane, we must determine whether to use an open or closed circle.
### Open vs. Closed Circle:
- **Open Circle**: Used when the endpoint is not included in the solution. This means the value exactly at that point is not part of the inequality.
- **Closed Circle**: Used when the endpoint is included in the solution. This means the value at that point is part of the inequality.
Since the inequality is \( n < -3 \) (not \( \leq \)), you would use an **open circle** at \(-3\) to indicate that \(-3\) is not included in the solution set.
### Direction:
For \( n < -3 \), the values of \( n \) are to the **left** of \(-3\) on a number line. This means you would shade or indicate the portion of the number line that extends leftward from the open circle at \(-3\).
### Summary:
- Open circle at \( -3 \)
- Shade or indicate to the left of \( -3 \)
### Open vs. Closed Circle:
- **Open Circle**: Used when the endpoint is not included in the solution. This means the value exactly at that point is not part of the inequality.
- **Closed Circle**: Used when the endpoint is included in the solution. This means the value at that point is part of the inequality.
Since the inequality is \( n < -3 \) (not \( \leq \)), you would use an **open circle** at \(-3\) to indicate that \(-3\) is not included in the solution set.
### Direction:
For \( n < -3 \), the values of \( n \) are to the **left** of \(-3\) on a number line. This means you would shade or indicate the portion of the number line that extends leftward from the open circle at \(-3\).
### Summary:
- Open circle at \( -3 \)
- Shade or indicate to the left of \( -3 \)
Answered by
GPT-4o mini
To solve the expression \( 2[32-(4-1)^3] \), we can follow the order of operations (parentheses, exponents, multiplication and division, addition and subtraction).
1. **Calculate the expression inside the parentheses**:
\[
4 - 1 = 3
\]
2. **Now raise it to the power of 3**:
\[
(4 - 1)^3 = 3^3 = 27
\]
3. **Substitute back into the expression**:
\[
2[32 - 27]
\]
4. **Perform the subtraction inside the brackets**:
\[
32 - 27 = 5
\]
5. **Now multiply by 2**:
\[
2 \times 5 = 10
\]
Thus, the value of the expression \( 2[32-(4-1)^3] \) is **10**.
1. **Calculate the expression inside the parentheses**:
\[
4 - 1 = 3
\]
2. **Now raise it to the power of 3**:
\[
(4 - 1)^3 = 3^3 = 27
\]
3. **Substitute back into the expression**:
\[
2[32 - 27]
\]
4. **Perform the subtraction inside the brackets**:
\[
32 - 27 = 5
\]
5. **Now multiply by 2**:
\[
2 \times 5 = 10
\]
Thus, the value of the expression \( 2[32-(4-1)^3] \) is **10**.
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