Question
y≥x+1 y≥5−x graphed out
Answers
GPT-4o mini
To graph the system of inequalities given by \(y \geq x + 1\) and \(y \geq 5 - x\), you can follow these steps:
### Step 1: Graph the equations
1. **Graph \(y = x + 1\)**:
- This is a straight line with a slope of 1 and a y-intercept of 1.
- It crosses the y-axis at (0, 1) and the x-axis at (-1, 0).
2. **Graph \(y = 5 - x\)**:
- This is another straight line with a slope of -1 and a y-intercept of 5.
- It crosses the y-axis at (0, 5) and the x-axis at (5, 0).
### Step 2: Determine the shading
Since both inequalities are of the form \(y \geq\), you will shade the region above both lines:
1. **For \(y \geq x + 1\)**: Shade above the line \(y = x + 1\).
2. **For \(y \geq 5 - x\)**: Shade above the line \(y = 5 - x\).
### Step 3: Find the intersection points
To determine where the two lines intersect, set \(x + 1 = 5 - x\):
\[
x + 1 = 5 - x
\]
\[
2x = 4
\]
\[
x = 2
\]
Now plug \(x = 2\) back into either equation to find \(y\):
\[
y = 2 + 1 = 3
\]
So the intersection point is \((2, 3)\).
### Step 4: Finalizing the graph
- Draw the two lines, using solid lines to indicate that points on the lines are included in the solution set.
- Shade the area above both lines, including the region that contains the intersection point \((2, 3)\) and area extending further upwards and right/left.
### Summary
The solution region consists of all points \((x, y)\) that lie above the lines \(y = x + 1\) and \(y = 5 - x\). The intersection point where both conditions meet is \((2, 3)\).
### Step 1: Graph the equations
1. **Graph \(y = x + 1\)**:
- This is a straight line with a slope of 1 and a y-intercept of 1.
- It crosses the y-axis at (0, 1) and the x-axis at (-1, 0).
2. **Graph \(y = 5 - x\)**:
- This is another straight line with a slope of -1 and a y-intercept of 5.
- It crosses the y-axis at (0, 5) and the x-axis at (5, 0).
### Step 2: Determine the shading
Since both inequalities are of the form \(y \geq\), you will shade the region above both lines:
1. **For \(y \geq x + 1\)**: Shade above the line \(y = x + 1\).
2. **For \(y \geq 5 - x\)**: Shade above the line \(y = 5 - x\).
### Step 3: Find the intersection points
To determine where the two lines intersect, set \(x + 1 = 5 - x\):
\[
x + 1 = 5 - x
\]
\[
2x = 4
\]
\[
x = 2
\]
Now plug \(x = 2\) back into either equation to find \(y\):
\[
y = 2 + 1 = 3
\]
So the intersection point is \((2, 3)\).
### Step 4: Finalizing the graph
- Draw the two lines, using solid lines to indicate that points on the lines are included in the solution set.
- Shade the area above both lines, including the region that contains the intersection point \((2, 3)\) and area extending further upwards and right/left.
### Summary
The solution region consists of all points \((x, y)\) that lie above the lines \(y = x + 1\) and \(y = 5 - x\). The intersection point where both conditions meet is \((2, 3)\).