Question
On a coordinate plane, a dashed straight line with positive slope goes through (negative 1.5, 0) and (0, 3). Everything to the right of the line is shaded.Which of the points are solutions to the inequality? Check all that apply. (–3, 3)(–2, –2)(–1, 1)(0, 1)(2, 5)
Answers
Answer
(–2, –2)
(0, 1)
(2, 5)
graph y<1/3x+1/2
(0, 1)
(2, 5)
graph y<1/3x+1/2
Answer
dot at 0,0.5 and then -1.5,0 the line has the lines like this - - - - - not ________ and shade on the bottom but Which point is a solution to the inequality?
(–2, 1)
(–1, 1)
(3, 1/2)
(–2, 1)
(–1, 1)
(3, 1/2)
Answer
A 2-column table has 4 rows. The first column is labeled x with entries negative 4, negative 2, 3, 3. The second column is labeled y with entries negative 1, 4, negative 3, negative 4.Which linear inequality could represent the given table of values?y < –2x + 3y ≤ –2x + 3y > –One-halfx – 3y ≤ –One-halfx – 3
Answer
Which statements are true about the graph of
y < Two-thirdsx + 1? Check all that apply.
The slope of the line is 1.
The line is solid.
The area below the line is shaded.
A solution to the inequality is (2, 3).
The x-intercept of the boundary line is (Negative three-halves, 0).
y < Two-thirdsx + 1? Check all that apply.
The slope of the line is 1.
The line is solid.
The area below the line is shaded.
A solution to the inequality is (2, 3).
The x-intercept of the boundary line is (Negative three-halves, 0).
Answer
Which is the graph of 2x + 3y > –3?On a coordinate plane, a dashed straight line with negative slope goes through (negative 2, 0) and (0, negative 1). Everything to the left of the line is shaded.On a coordinate plane, a dashed straight line with negative slope goes through (negative 2, 2) and (0, negative 1). Everything to the right of the line is shaded.On a coordinate plane, a dashed straight line with negative slope goes through (negative 2, 0) and (0, negative 1). Everything to the right of the line is shaded.On a coordinate plane, a dashed straight line with negative slope goes through (negative 2, 2) and (0, negative 1). Everything to the left of the line is shaded.
Answered by
GPT-4o mini
To determine which of the points are solutions to the inequality represented by the dashed line going through the points (-1.5, 0) and (0, 3), we first need to find the equation of this line.
1. **Finding the slope of the line:**
The slope \( m \) of a line can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points (0, 3) (which we can accept as \( (x_2,y_2) \)) and (-1.5, 0) (which we accept as \( (x_1,y_1) \)):
\[
m = \frac{3 - 0}{0 - (-1.5)} = \frac{3}{1.5} = 2
\]
2. **Finding the y-intercept:**
The y-intercept from the point (0, 3) is \( 3 \).
3. **Equation of the line:**
We can use the point-slope form based on the slope \( m = 2 \) and y-intercept \( b = 3 \):
\[
y = 2x + 3
\]
Since the line is dashed, the inequality that represents the area to the right of the line is:
\[
y < 2x + 3
\]
4. **Testing each point:**
- **For (-3, 3):**
\[
3 < 2(-3) + 3 \Longrightarrow 3 < -6 + 3 \Longrightarrow 3 < -3 \quad \text{(False)}
\]
- **For (-2, -2):**
\[
-2 < 2(-2) + 3 \Longrightarrow -2 < -4 + 3 \Longrightarrow -2 < -1 \quad \text{(False)}
\]
- **For (-1, 1):**
\[
1 < 2(-1) + 3 \Longrightarrow 1 < -2 + 3 \Longrightarrow 1 < 1 \quad \text{(False)}
\]
- **For (0, 1):**
\[
1 < 2(0) + 3 \Longrightarrow 1 < 0 + 3 \Longrightarrow 1 < 3 \quad \text{(True)}
\]
- **For (2, 5):**
\[
5 < 2(2) + 3 \Longrightarrow 5 < 4 + 3 \Longrightarrow 5 < 7 \quad \text{(True)}
\]
5. **Summary of results:**
The points that are solutions to the inequality \( y < 2x + 3 \) are:
- (0, 1)
- (2, 5)
Thus, the correct answers are:
- (0, 1)
- (2, 5)
1. **Finding the slope of the line:**
The slope \( m \) of a line can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points (0, 3) (which we can accept as \( (x_2,y_2) \)) and (-1.5, 0) (which we accept as \( (x_1,y_1) \)):
\[
m = \frac{3 - 0}{0 - (-1.5)} = \frac{3}{1.5} = 2
\]
2. **Finding the y-intercept:**
The y-intercept from the point (0, 3) is \( 3 \).
3. **Equation of the line:**
We can use the point-slope form based on the slope \( m = 2 \) and y-intercept \( b = 3 \):
\[
y = 2x + 3
\]
Since the line is dashed, the inequality that represents the area to the right of the line is:
\[
y < 2x + 3
\]
4. **Testing each point:**
- **For (-3, 3):**
\[
3 < 2(-3) + 3 \Longrightarrow 3 < -6 + 3 \Longrightarrow 3 < -3 \quad \text{(False)}
\]
- **For (-2, -2):**
\[
-2 < 2(-2) + 3 \Longrightarrow -2 < -4 + 3 \Longrightarrow -2 < -1 \quad \text{(False)}
\]
- **For (-1, 1):**
\[
1 < 2(-1) + 3 \Longrightarrow 1 < -2 + 3 \Longrightarrow 1 < 1 \quad \text{(False)}
\]
- **For (0, 1):**
\[
1 < 2(0) + 3 \Longrightarrow 1 < 0 + 3 \Longrightarrow 1 < 3 \quad \text{(True)}
\]
- **For (2, 5):**
\[
5 < 2(2) + 3 \Longrightarrow 5 < 4 + 3 \Longrightarrow 5 < 7 \quad \text{(True)}
\]
5. **Summary of results:**
The points that are solutions to the inequality \( y < 2x + 3 \) are:
- (0, 1)
- (2, 5)
Thus, the correct answers are:
- (0, 1)
- (2, 5)
Answered by
GPT-4o mini
To graph the inequality \(y < \frac{1}{3}x + \frac{1}{2}\), follow these steps:
### Step 1: Graph the Line
1. **Identify the slope and y-intercept:**
The equation is in slope-intercept form \(y = mx + b\), where:
- Slope (\(m\)) = \( \frac{1}{3} \)
- y-intercept (\(b\)) = \( \frac{1}{2} \)
2. **Plot the y-intercept:**
Begin by plotting the point \((0, \frac{1}{2})\) on the coordinate plane. This is the point where the line crosses the y-axis.
3. **Use the slope to find another point:**
The slope of \( \frac{1}{3} \) means that for every 3 units you move to the right (positive x-direction), you move 1 unit up (positive y-direction).
- Starting from \((0, \frac{1}{2})\), if you move 3 units right to \((3, \frac{1}{2})\), then go up 1 unit to \((3, \frac{1}{2} + 1)\), which gives us \((3, \frac{3}{2})\).
4. **Draw the line:**
Connect the points \((0, \frac{1}{2})\) and \((3, \frac{3}{2})\) with a dashed line because the inequality is strict (\(<\)), indicating that points on the line itself are not included in the solution set.
### Step 2: Shade the Region for the Inequality
Since the inequality is \(y < \frac{1}{3}x + \frac{1}{2}\), you will shade the area below the line.
- Choose a test point that is not on the line (a good choice is usually (0, 0)):
\[
0 < \frac{1}{3}(0) + \frac{1}{2} \Longrightarrow 0 < \frac{1}{2} \quad \text{(True)}
\]
- This confirms that (0, 0) is a solution, so you shade the region below the line.
### Summary of the Graph
- You have a dashed line (indicating that points on the line are not included in the solution set) that crosses the y-axis at \((0, \frac{1}{2})\) and has a slope of \(\frac{1}{3}\).
- Shade below this line to represent all points \((x, y)\) that satisfy the inequality \(y < \frac{1}{3}x + \frac{1}{2}\).
### Final Graph
- The graph will visually represent a dashed line with the shaded region below it.
- If you are using graph paper or graphing software, you can mark the points you calculated and follow the instructions above to complete the graph accurately.
### Step 1: Graph the Line
1. **Identify the slope and y-intercept:**
The equation is in slope-intercept form \(y = mx + b\), where:
- Slope (\(m\)) = \( \frac{1}{3} \)
- y-intercept (\(b\)) = \( \frac{1}{2} \)
2. **Plot the y-intercept:**
Begin by plotting the point \((0, \frac{1}{2})\) on the coordinate plane. This is the point where the line crosses the y-axis.
3. **Use the slope to find another point:**
The slope of \( \frac{1}{3} \) means that for every 3 units you move to the right (positive x-direction), you move 1 unit up (positive y-direction).
- Starting from \((0, \frac{1}{2})\), if you move 3 units right to \((3, \frac{1}{2})\), then go up 1 unit to \((3, \frac{1}{2} + 1)\), which gives us \((3, \frac{3}{2})\).
4. **Draw the line:**
Connect the points \((0, \frac{1}{2})\) and \((3, \frac{3}{2})\) with a dashed line because the inequality is strict (\(<\)), indicating that points on the line itself are not included in the solution set.
### Step 2: Shade the Region for the Inequality
Since the inequality is \(y < \frac{1}{3}x + \frac{1}{2}\), you will shade the area below the line.
- Choose a test point that is not on the line (a good choice is usually (0, 0)):
\[
0 < \frac{1}{3}(0) + \frac{1}{2} \Longrightarrow 0 < \frac{1}{2} \quad \text{(True)}
\]
- This confirms that (0, 0) is a solution, so you shade the region below the line.
### Summary of the Graph
- You have a dashed line (indicating that points on the line are not included in the solution set) that crosses the y-axis at \((0, \frac{1}{2})\) and has a slope of \(\frac{1}{3}\).
- Shade below this line to represent all points \((x, y)\) that satisfy the inequality \(y < \frac{1}{3}x + \frac{1}{2}\).
### Final Graph
- The graph will visually represent a dashed line with the shaded region below it.
- If you are using graph paper or graphing software, you can mark the points you calculated and follow the instructions above to complete the graph accurately.
Answered by
GPT-4o mini
To determine which of the given points is a solution to the inequality \(y < \frac{1}{3}x + \frac{1}{2}\), we can substitute each point into the inequality and check if it holds true.
### Inequality Reiteration
The inequality we are examining is:
\[
y < \frac{1}{3}x + \frac{1}{2}
\]
### Test Each Point
1. **For Point \((-2, 1)\):**
- Substitute \(x = -2\) and \(y = 1\):
\[
1 < \frac{1}{3}(-2) + \frac{1}{2}
\]
Simplifying the right side:
\[
1 < \left(-\frac{2}{3}\right) + \frac{1}{2}
\]
Converting \(\frac{1}{2}\) to a fraction with a common denominator:
\[
\frac{1}{2} = \frac{3}{6} = \frac{9}{3}
\]
Therefore,
\[
1 < -\frac{2}{3} + \frac{3}{6} = \frac{-4 + 3}{6} = -\frac{1}{6}
\]
Since \(1 < -\frac{1}{6}\) is **False**, \((-2, 1)\) is **not** a solution.
2. **For Point \((-1, 1)\):**
- Substitute \(x = -1\) and \(y = 1\):
\[
1 < \frac{1}{3}(-1) + \frac{1}{2}
\]
Simplifying the right side:
\[
1 < -\frac{1}{3} + \frac{1}{2}
\]
Converting \(\frac{1}{2}\) to fractions with a common denominator:
\[
\frac{1}{2} = \frac{3}{6} \quad \text{and} \quad -\frac{1}{3} = \frac{-2}{6}
\]
So,
\[
1 < -\frac{2}{6} + \frac{3}{6} = \frac{1}{6}
\]
Since \(1 < \frac{1}{6}\) is **False**, \((-1, 1)\) is **not** a solution.
3. **For Point \((3, \frac{1}{2})\):**
- Substitute \(x = 3\) and \(y = \frac{1}{2}\):
\[
\frac{1}{2} < \frac{1}{3}(3) + \frac{1}{2}
\]
Simplifying the right side:
\[
\frac{1}{2} < 1 + \frac{1}{2}
\]
Therefore,
\[
\frac{1}{2} < \frac{2}{2} + \frac{1}{2} = \frac{3}{2}
\]
Since \(\frac{1}{2} < \frac{3}{2}\) is **True**, \((3, \frac{1}{2})\) **is** a solution.
### Conclusion
The only solution to the inequality \(y < \frac{1}{3}x + \frac{1}{2}\) from the given points is:
- **(3, \(\frac{1}{2}\))**.
### Inequality Reiteration
The inequality we are examining is:
\[
y < \frac{1}{3}x + \frac{1}{2}
\]
### Test Each Point
1. **For Point \((-2, 1)\):**
- Substitute \(x = -2\) and \(y = 1\):
\[
1 < \frac{1}{3}(-2) + \frac{1}{2}
\]
Simplifying the right side:
\[
1 < \left(-\frac{2}{3}\right) + \frac{1}{2}
\]
Converting \(\frac{1}{2}\) to a fraction with a common denominator:
\[
\frac{1}{2} = \frac{3}{6} = \frac{9}{3}
\]
Therefore,
\[
1 < -\frac{2}{3} + \frac{3}{6} = \frac{-4 + 3}{6} = -\frac{1}{6}
\]
Since \(1 < -\frac{1}{6}\) is **False**, \((-2, 1)\) is **not** a solution.
2. **For Point \((-1, 1)\):**
- Substitute \(x = -1\) and \(y = 1\):
\[
1 < \frac{1}{3}(-1) + \frac{1}{2}
\]
Simplifying the right side:
\[
1 < -\frac{1}{3} + \frac{1}{2}
\]
Converting \(\frac{1}{2}\) to fractions with a common denominator:
\[
\frac{1}{2} = \frac{3}{6} \quad \text{and} \quad -\frac{1}{3} = \frac{-2}{6}
\]
So,
\[
1 < -\frac{2}{6} + \frac{3}{6} = \frac{1}{6}
\]
Since \(1 < \frac{1}{6}\) is **False**, \((-1, 1)\) is **not** a solution.
3. **For Point \((3, \frac{1}{2})\):**
- Substitute \(x = 3\) and \(y = \frac{1}{2}\):
\[
\frac{1}{2} < \frac{1}{3}(3) + \frac{1}{2}
\]
Simplifying the right side:
\[
\frac{1}{2} < 1 + \frac{1}{2}
\]
Therefore,
\[
\frac{1}{2} < \frac{2}{2} + \frac{1}{2} = \frac{3}{2}
\]
Since \(\frac{1}{2} < \frac{3}{2}\) is **True**, \((3, \frac{1}{2})\) **is** a solution.
### Conclusion
The only solution to the inequality \(y < \frac{1}{3}x + \frac{1}{2}\) from the given points is:
- **(3, \(\frac{1}{2}\))**.
Answered by
GPT-4o mini
To determine which linear inequality best represents the given table of values, we will analyze the points in the table and check each provided inequality.
### Given Points
The points in the table are:
1. (-4, -1)
2. (-2, 4)
3. (3, -3)
4. (3, -4)
### Check Each Inequality
#### **1. Inequality: \(y < -2x + 3\)**
We will evaluate this inequality against all points.
- **For (-4, -1):**
\[
-1 < -2(-4) + 3 \Rightarrow -1 < 8 + 3 \Rightarrow -1 < 11 \quad \text{(True)}
\]
- **For (-2, 4):**
\[
4 < -2(-2) + 3 \Rightarrow 4 < 4 + 3 \Rightarrow 4 < 7 \quad \text{(True)}
\]
- **For (3, -3):**
\[
-3 < -2(3) + 3 \Rightarrow -3 < -6 + 3 \Rightarrow -3 < -3 \quad \text{(False)}
\]
- **For (3, -4):**
\[
-4 < -2(3) + 3 \Rightarrow -4 < -6 + 3 \Rightarrow -4 < -3 \quad \text{(False)}
\]
**Conclusion for this inequality:** It does not hold for all points.
#### **2. Inequality: \(y ≤ -2x + 3\)**
- **For (-4, -1):**
\[
-1 ≤ -2(-4) + 3 \Rightarrow -1 ≤ 8 + 3 \Rightarrow -1 ≤ 11 \quad \text{(True)}
\]
- **For (-2, 4):**
\[
4 ≤ -2(-2) + 3 \Rightarrow 4 ≤ 4 + 3 \Rightarrow 4 ≤ 7 \quad \text{(True)}
\]
- **For (3, -3):**
\[
-3 ≤ -2(3) + 3 \Rightarrow -3 ≤ -6 + 3 \Rightarrow -3 ≤ -3 \quad \text{(True)}
\]
- **For (3, -4):**
\[
-4 ≤ -2(3) + 3 \Rightarrow -4 ≤ -6 + 3 \Rightarrow -4 ≤ -3 \quad \text{(False)}
\]
**Conclusion for this inequality:** It does not hold for all points.
#### **3. Inequality: \(y > -\frac{1}{2}x - 3\)**
- **For (-4, -1):**
\[
-1 > -\frac{1}{2}(-4) - 3 \Rightarrow -1 > 2 - 3 \Rightarrow -1 > -1 \quad \text{(False)}
\]
- **For (-2, 4):**
\[
4 > -\frac{1}{2}(-2) - 3 \Rightarrow 4 > 1 - 3 \Rightarrow 4 > -2 \quad \text{(True)}
\]
- **For (3, -3):**
\[
-3 > -\frac{1}{2}(3) - 3 \Rightarrow -3 > -\frac{3}{2} - 3 \Rightarrow -3 > -\frac{3}{2} - \frac{6}{2} \Rightarrow -3 > -\frac{9}{2} \text{ (True)}
\]
- **For (3, -4):**
\[
-4 > -\frac{1}{2}(3) - 3 \Rightarrow -4 > -\frac{3}{2} - 3 \Rightarrow -4 > -\frac{3}{2} - \frac{6}{2} \Rightarrow -4 > -\frac{9}{2} \text{ (True)}
\]
**Conclusion for this inequality:** It does not hold for all points.
#### **4. Inequality: \(y ≤ -\frac{1}{2}x - 3\)**
- **For (-4, -1):**
\[
-1 ≤ -\frac{1}{2}(-4)-3 \Rightarrow -1 ≤ 2 - 3 \Rightarrow -1 ≤ -1 \quad \text{(True)}
\]
- **For (-2, 4):**
\[
4 ≤ -\frac{1}{2}(-2)-3 \Rightarrow 4 ≤ 1 - 3 \Rightarrow 4 ≤ -2 \quad \text{(False)}
\]
- **For (3, -3):**
\[
-3 ≤ -\frac{1}{2}(3)-3 \Rightarrow -3 ≤ -\frac{3}{2} - 3 \Rightarrow -3 ≤ -\frac{3}{2} - \frac{6}{2} \Rightarrow -3 ≤ -\frac{9}{2} \quad \text{(True)}
\]
- **For (3, -4):**
\[
-4 ≤ -\frac{1}{2}(3)-3 \Rightarrow -4 ≤ -\frac{3}{2} - 3 \Rightarrow -4 ≤ -\frac{3}{2} - \frac{6}{2} \Rightarrow -4 ≤ -\frac{9}{2} \quad \text{(True)}
\]
**Conclusion for this inequality:** It does not hold for all points.
### Summary of Findings
Based on the evaluations, none of the given inequalities hold for all the points in the table. However, the most appropriate conclusion would be to consider that the second inequality \(y \leq -2x + 3\) holds true for three out of the four points but still fails for one point. Hence, it might be the best fit among given options if we were to prioritize the most frequent true evaluations.
Given that, the inequality that could represent the points is:
**y ≤ -2x + 3**, although it might not strictly satisfy all points perfectly. If none of the inequalities seem to satisfy all points, then we might consider verifying the points further or revising the search for a better fitting inequality.
### Given Points
The points in the table are:
1. (-4, -1)
2. (-2, 4)
3. (3, -3)
4. (3, -4)
### Check Each Inequality
#### **1. Inequality: \(y < -2x + 3\)**
We will evaluate this inequality against all points.
- **For (-4, -1):**
\[
-1 < -2(-4) + 3 \Rightarrow -1 < 8 + 3 \Rightarrow -1 < 11 \quad \text{(True)}
\]
- **For (-2, 4):**
\[
4 < -2(-2) + 3 \Rightarrow 4 < 4 + 3 \Rightarrow 4 < 7 \quad \text{(True)}
\]
- **For (3, -3):**
\[
-3 < -2(3) + 3 \Rightarrow -3 < -6 + 3 \Rightarrow -3 < -3 \quad \text{(False)}
\]
- **For (3, -4):**
\[
-4 < -2(3) + 3 \Rightarrow -4 < -6 + 3 \Rightarrow -4 < -3 \quad \text{(False)}
\]
**Conclusion for this inequality:** It does not hold for all points.
#### **2. Inequality: \(y ≤ -2x + 3\)**
- **For (-4, -1):**
\[
-1 ≤ -2(-4) + 3 \Rightarrow -1 ≤ 8 + 3 \Rightarrow -1 ≤ 11 \quad \text{(True)}
\]
- **For (-2, 4):**
\[
4 ≤ -2(-2) + 3 \Rightarrow 4 ≤ 4 + 3 \Rightarrow 4 ≤ 7 \quad \text{(True)}
\]
- **For (3, -3):**
\[
-3 ≤ -2(3) + 3 \Rightarrow -3 ≤ -6 + 3 \Rightarrow -3 ≤ -3 \quad \text{(True)}
\]
- **For (3, -4):**
\[
-4 ≤ -2(3) + 3 \Rightarrow -4 ≤ -6 + 3 \Rightarrow -4 ≤ -3 \quad \text{(False)}
\]
**Conclusion for this inequality:** It does not hold for all points.
#### **3. Inequality: \(y > -\frac{1}{2}x - 3\)**
- **For (-4, -1):**
\[
-1 > -\frac{1}{2}(-4) - 3 \Rightarrow -1 > 2 - 3 \Rightarrow -1 > -1 \quad \text{(False)}
\]
- **For (-2, 4):**
\[
4 > -\frac{1}{2}(-2) - 3 \Rightarrow 4 > 1 - 3 \Rightarrow 4 > -2 \quad \text{(True)}
\]
- **For (3, -3):**
\[
-3 > -\frac{1}{2}(3) - 3 \Rightarrow -3 > -\frac{3}{2} - 3 \Rightarrow -3 > -\frac{3}{2} - \frac{6}{2} \Rightarrow -3 > -\frac{9}{2} \text{ (True)}
\]
- **For (3, -4):**
\[
-4 > -\frac{1}{2}(3) - 3 \Rightarrow -4 > -\frac{3}{2} - 3 \Rightarrow -4 > -\frac{3}{2} - \frac{6}{2} \Rightarrow -4 > -\frac{9}{2} \text{ (True)}
\]
**Conclusion for this inequality:** It does not hold for all points.
#### **4. Inequality: \(y ≤ -\frac{1}{2}x - 3\)**
- **For (-4, -1):**
\[
-1 ≤ -\frac{1}{2}(-4)-3 \Rightarrow -1 ≤ 2 - 3 \Rightarrow -1 ≤ -1 \quad \text{(True)}
\]
- **For (-2, 4):**
\[
4 ≤ -\frac{1}{2}(-2)-3 \Rightarrow 4 ≤ 1 - 3 \Rightarrow 4 ≤ -2 \quad \text{(False)}
\]
- **For (3, -3):**
\[
-3 ≤ -\frac{1}{2}(3)-3 \Rightarrow -3 ≤ -\frac{3}{2} - 3 \Rightarrow -3 ≤ -\frac{3}{2} - \frac{6}{2} \Rightarrow -3 ≤ -\frac{9}{2} \quad \text{(True)}
\]
- **For (3, -4):**
\[
-4 ≤ -\frac{1}{2}(3)-3 \Rightarrow -4 ≤ -\frac{3}{2} - 3 \Rightarrow -4 ≤ -\frac{3}{2} - \frac{6}{2} \Rightarrow -4 ≤ -\frac{9}{2} \quad \text{(True)}
\]
**Conclusion for this inequality:** It does not hold for all points.
### Summary of Findings
Based on the evaluations, none of the given inequalities hold for all the points in the table. However, the most appropriate conclusion would be to consider that the second inequality \(y \leq -2x + 3\) holds true for three out of the four points but still fails for one point. Hence, it might be the best fit among given options if we were to prioritize the most frequent true evaluations.
Given that, the inequality that could represent the points is:
**y ≤ -2x + 3**, although it might not strictly satisfy all points perfectly. If none of the inequalities seem to satisfy all points, then we might consider verifying the points further or revising the search for a better fitting inequality.
Answered by
GPT-4o mini
To analyze the inequality \(y < \frac{2}{3}x + 1\) and determine the accuracy of each statement, let’s break down the characteristics of the graph and the relevant calculations:
### 1. **The slope of the line is 1.**
- **False.** The slope of the line given by the equation \(y = \frac{2}{3}x + 1\) is \(\frac{2}{3}\), not 1.
### 2. **The line is solid.**
- **False.** Since the inequality is \(y < \frac{2}{3}x + 1\) (strict inequality), the line representing the equation \(y = \frac{2}{3}x + 1\) is dashed.
### 3. **The area below the line is shaded.**
- **True.** The inequality \(y < \frac{2}{3}x + 1\) indicates that we shade the area below the line because the solutions to the inequality are all the points where \(y\) is less than the value of the line at any given \(x\).
### 4. **A solution to the inequality is (2, 3).**
- **False.** To check if the point (2, 3) is a solution, substitute \(x = 2\) into the inequality:
\[
3 < \frac{2}{3}(2) + 1 \Rightarrow 3 < \frac{4}{3} + 1 \Rightarrow 3 < \frac{4}{3} + \frac{3}{3} = \frac{7}{3} \quad \text{(False)}
\]
The point (2, 3) does not satisfy the inequality.
### 5. **The x-intercept of the boundary line is \((-3/2, 0)\).**
- **True.** To find the x-intercept, set \(y = 0\) in the equation \(y = \frac{2}{3}x + 1\):
\[
0 = \frac{2}{3}x + 1 \Rightarrow -1 = \frac{2}{3}x \Rightarrow x = -\frac{3}{2}
\]
Thus, the x-intercept is indeed \((-3/2, 0)\).
### Summary of True Statements
The true statements about the graph of \(y < \frac{2}{3}x + 1\) are:
- **The area below the line is shaded.**
- **The x-intercept of the boundary line is \((-3/2, 0)\).**
Therefore, while the other statements are false, the correct truths regarding the inequality are as stated above.
### 1. **The slope of the line is 1.**
- **False.** The slope of the line given by the equation \(y = \frac{2}{3}x + 1\) is \(\frac{2}{3}\), not 1.
### 2. **The line is solid.**
- **False.** Since the inequality is \(y < \frac{2}{3}x + 1\) (strict inequality), the line representing the equation \(y = \frac{2}{3}x + 1\) is dashed.
### 3. **The area below the line is shaded.**
- **True.** The inequality \(y < \frac{2}{3}x + 1\) indicates that we shade the area below the line because the solutions to the inequality are all the points where \(y\) is less than the value of the line at any given \(x\).
### 4. **A solution to the inequality is (2, 3).**
- **False.** To check if the point (2, 3) is a solution, substitute \(x = 2\) into the inequality:
\[
3 < \frac{2}{3}(2) + 1 \Rightarrow 3 < \frac{4}{3} + 1 \Rightarrow 3 < \frac{4}{3} + \frac{3}{3} = \frac{7}{3} \quad \text{(False)}
\]
The point (2, 3) does not satisfy the inequality.
### 5. **The x-intercept of the boundary line is \((-3/2, 0)\).**
- **True.** To find the x-intercept, set \(y = 0\) in the equation \(y = \frac{2}{3}x + 1\):
\[
0 = \frac{2}{3}x + 1 \Rightarrow -1 = \frac{2}{3}x \Rightarrow x = -\frac{3}{2}
\]
Thus, the x-intercept is indeed \((-3/2, 0)\).
### Summary of True Statements
The true statements about the graph of \(y < \frac{2}{3}x + 1\) are:
- **The area below the line is shaded.**
- **The x-intercept of the boundary line is \((-3/2, 0)\).**
Therefore, while the other statements are false, the correct truths regarding the inequality are as stated above.
Answered by
GPT-4o mini
To determine which graph represents the inequality \(2x + 3y > -3\), we need to first rewrite it in slope-intercept form (i.e., \(y = mx + b\)) and analyze its components.
### Step 1: Rewrite the inequality
1. Start with the inequality:
\[
2x + 3y > -3
\]
2. Isolate \(y\):
\[
3y > -2x - 3
\]
\[
y > -\frac{2}{3}x - 1
\]
### Step 2: Determine the slope and y-intercept
- The slope (\(m\)) is \(-\frac{2}{3}\) (negative slope).
- The y-intercept (\(b\)) is \(-1\) (the point at which the line crosses the y-axis is (0, -1)).
### Step 3: Identify the x-intercept
To identify additional information, calculate the x-intercept by setting \(y = 0\):
\[
0 = -\frac{2}{3}x - 1 \implies -1 = -\frac{2}{3}x \implies x = \frac{3}{2} \text{ or } (1.5, 0)
\]
### Step 4: Characteristics of the line
- The line that corresponds to the equation will be dashed since the inequality is strict (\(>\)).
- The region above the line (because \(y >\)) will be shaded.
### Step 5: Analyze the options
Now, we examine the provided graph options based on the characteristics above:
1. **Option 1:** A dashed straight line with negative slope going through \((-2, 0)\) and \((0, -1)\), with shading to the left of the line.
- **Incorrect**, since we should shade above the line.
2. **Option 2:** A dashed straight line with negative slope going through \((-2, 2)\) and \((0, -1)\), with shading to the right of the line.
- **Incorrect**, since it does not go through the correct points and shading should be above.
3. **Option 3:** A dashed straight line with negative slope going through \((-2, 0)\) and \((0, -1)\), with shading to the right of the line.
- **Incorrect**, shading should be above, not to the right.
4. **Option 4:** A dashed straight line with negative slope going through \((-2, 2)\) and \((0, -1)\), with shading to the left of the line.
- **Incorrect**, since shading should be above the line, not to the left.
### Conclusion
None of the provided descriptions of the graphs correctly match the graph of the inequality \(2x + 3y > -3\) (or \(y > -\frac{2}{3}x - 1\)).
For this inequality to be accurately represented:
- The line would start at \((0, -1)\) with a negative slope extending downwards (as described).
- The shading should be above the line.
If we were to accurately represent \(2x + 3y > -3\), we would need a graph with the line running correctly as per calculated intercept points and shading above the dashed line. If you have any specific visual options or graphs, I can help you analyze them further.
### Step 1: Rewrite the inequality
1. Start with the inequality:
\[
2x + 3y > -3
\]
2. Isolate \(y\):
\[
3y > -2x - 3
\]
\[
y > -\frac{2}{3}x - 1
\]
### Step 2: Determine the slope and y-intercept
- The slope (\(m\)) is \(-\frac{2}{3}\) (negative slope).
- The y-intercept (\(b\)) is \(-1\) (the point at which the line crosses the y-axis is (0, -1)).
### Step 3: Identify the x-intercept
To identify additional information, calculate the x-intercept by setting \(y = 0\):
\[
0 = -\frac{2}{3}x - 1 \implies -1 = -\frac{2}{3}x \implies x = \frac{3}{2} \text{ or } (1.5, 0)
\]
### Step 4: Characteristics of the line
- The line that corresponds to the equation will be dashed since the inequality is strict (\(>\)).
- The region above the line (because \(y >\)) will be shaded.
### Step 5: Analyze the options
Now, we examine the provided graph options based on the characteristics above:
1. **Option 1:** A dashed straight line with negative slope going through \((-2, 0)\) and \((0, -1)\), with shading to the left of the line.
- **Incorrect**, since we should shade above the line.
2. **Option 2:** A dashed straight line with negative slope going through \((-2, 2)\) and \((0, -1)\), with shading to the right of the line.
- **Incorrect**, since it does not go through the correct points and shading should be above.
3. **Option 3:** A dashed straight line with negative slope going through \((-2, 0)\) and \((0, -1)\), with shading to the right of the line.
- **Incorrect**, shading should be above, not to the right.
4. **Option 4:** A dashed straight line with negative slope going through \((-2, 2)\) and \((0, -1)\), with shading to the left of the line.
- **Incorrect**, since shading should be above the line, not to the left.
### Conclusion
None of the provided descriptions of the graphs correctly match the graph of the inequality \(2x + 3y > -3\) (or \(y > -\frac{2}{3}x - 1\)).
For this inequality to be accurately represented:
- The line would start at \((0, -1)\) with a negative slope extending downwards (as described).
- The shading should be above the line.
If we were to accurately represent \(2x + 3y > -3\), we would need a graph with the line running correctly as per calculated intercept points and shading above the dashed line. If you have any specific visual options or graphs, I can help you analyze them further.
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