Question
Which of the following inequalities is `(2,2)` a solution?
y<-2x+1
y\ge3x-2
y>x-1
y<-2x+1
y\ge3x-2
y>x-1
Answers
Answer
Graph the inequality.
`y<x+5`
`y<x+5`
Answered by
GPT-4o mini
To determine which inequalities the point (2, 2) is a solution for, we will substitute \(x = 2\) and \(y = 2\) into each inequality and check if the inequality holds.
1. **For the inequality \(y < -2x + 1\)**:
Substitute \(x = 2\) and \(y = 2\):
\[
2 < -2(2) + 1 \implies 2 < -4 + 1 \implies 2 < -3
\]
This is **false**, so (2, 2) is not a solution for this inequality.
2. **For the inequality \(y \geq 3x - 2\)**:
Substitute \(x = 2\) and \(y = 2\):
\[
2 \geq 3(2) - 2 \implies 2 \geq 6 - 2 \implies 2 \geq 4
\]
This is **false**, so (2, 2) is not a solution for this inequality.
3. **For the inequality \(y > x - 1\)**:
Substitute \(x = 2\) and \(y = 2\):
\[
2 > 2 - 1 \implies 2 > 1
\]
This is **true**, so (2, 2) is a solution for this inequality.
In summary, (2, 2) is only a solution for the inequality \(y > x - 1\).
1. **For the inequality \(y < -2x + 1\)**:
Substitute \(x = 2\) and \(y = 2\):
\[
2 < -2(2) + 1 \implies 2 < -4 + 1 \implies 2 < -3
\]
This is **false**, so (2, 2) is not a solution for this inequality.
2. **For the inequality \(y \geq 3x - 2\)**:
Substitute \(x = 2\) and \(y = 2\):
\[
2 \geq 3(2) - 2 \implies 2 \geq 6 - 2 \implies 2 \geq 4
\]
This is **false**, so (2, 2) is not a solution for this inequality.
3. **For the inequality \(y > x - 1\)**:
Substitute \(x = 2\) and \(y = 2\):
\[
2 > 2 - 1 \implies 2 > 1
\]
This is **true**, so (2, 2) is a solution for this inequality.
In summary, (2, 2) is only a solution for the inequality \(y > x - 1\).
Answered by
GPT-4o mini
To graph the inequality \( y < x + 5 \), follow these steps:
1. **Graph the boundary line**:
- Start by graphing the equation \( y = x + 5 \).
- This is a straight line with a slope of 1 (rise over run) and a y-intercept of 5. This means it crosses the y-axis at (0, 5).
To plot the line:
- Plot the y-intercept (0, 5).
- Choose another point by selecting a value for \( x \). For example, if \( x = 0 \), then \( y = 5 \); if \( x = 1 \), then \( y = 6 \) (this gives the point (1, 6)).
- Connect these points with a straight line.
2. **Dashed line**:
- Since the inequality is \( y < x + 5 \) (and not \( y \leq x + 5 \)), draw the line as a **dashed line**. This indicates that points on the line are not included in the solution.
3. **Shading the region**:
- Since the inequality is \( y < x + 5\), you will shade **below** the line, which represents all the points where \( y \) is less than the line \( y = x + 5 \).
**Final Remarks**:
- The graph will show a dashed line representing the boundary \( y = x + 5 \) with the region below it shaded to indicate the solutions to the inequality.
You can use graphing software or graph paper to visualize it based on the description provided.
1. **Graph the boundary line**:
- Start by graphing the equation \( y = x + 5 \).
- This is a straight line with a slope of 1 (rise over run) and a y-intercept of 5. This means it crosses the y-axis at (0, 5).
To plot the line:
- Plot the y-intercept (0, 5).
- Choose another point by selecting a value for \( x \). For example, if \( x = 0 \), then \( y = 5 \); if \( x = 1 \), then \( y = 6 \) (this gives the point (1, 6)).
- Connect these points with a straight line.
2. **Dashed line**:
- Since the inequality is \( y < x + 5 \) (and not \( y \leq x + 5 \)), draw the line as a **dashed line**. This indicates that points on the line are not included in the solution.
3. **Shading the region**:
- Since the inequality is \( y < x + 5\), you will shade **below** the line, which represents all the points where \( y \) is less than the line \( y = x + 5 \).
**Final Remarks**:
- The graph will show a dashed line representing the boundary \( y = x + 5 \) with the region below it shaded to indicate the solutions to the inequality.
You can use graphing software or graph paper to visualize it based on the description provided.
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