Answer :
Sure, let's go step by step to graph the inequality [tex]\( -x + 4y < 12 \)[/tex].
### Step 1: Rewrite the inequality as an equation
First, we convert the inequality into an equation to find the boundary line:
[tex]\[ -x + 4y = 12 \][/tex]
### Step 2: Solve for [tex]\( y \)[/tex] to get the equation in slope-intercept form
Solving for [tex]\( y \)[/tex]:
[tex]\[ 4y = x + 12 \][/tex]
[tex]\[ y = \frac{1}{4}x + 3 \][/tex]
This tells us that the line has a slope of [tex]\( \frac{1}{4} \)[/tex] and a y-intercept of 3.
### Step 3: Plot the boundary line
We plot the line [tex]\( y = \frac{1}{4}x + 3 \)[/tex]. This line will divide the plane into two regions.
### Step 4: Determine the shading (inequality region)
We need to determine which side of the line to shade for the inequality [tex]\( -x + 4y < 12 \)[/tex].
Choose a test point that is not on the line; the origin [tex]\((0,0)\)[/tex] is often a convenient choice. Substitute [tex]\((0,0)\)[/tex] into the inequality:
[tex]\[ -0 + 4(0) < 12 \][/tex]
[tex]\[ 0 < 12 \][/tex]
This is true, so the region that includes the origin is the solution to the inequality. This means we shade the region below the line.
### Step 5: Draw the graph
1. Plot the line [tex]\( y = \frac{1}{4}x + 3 \)[/tex]:
- The y-intercept is at [tex]\( (0, 3) \)[/tex].
- The slope [tex]\( \frac{1}{4} \)[/tex] means that for every increase of 1 unit in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\( \frac{1}{4} \)[/tex] units. Therefore, you can plot another point by moving 4 units to the right (positive [tex]\( x \)[/tex]) and 1 unit up (positive [tex]\( y \)[/tex]).
- Two points to plot could be [tex]\( (0, 3) \)[/tex] and [tex]\( (4, 4) \)[/tex], then draw the line through these points.
2. Shade the region below the line:
- As determined earlier, the region below this line (including the test point (0,0)) will be shaded to represent [tex]\( -x + 4y < 12 \)[/tex].
### Sketch of the graph:
- Draw the line with points like [tex]\((0,3)\)[/tex] and [tex]\((4,4)\)[/tex].
- Dashed lines usually represent the boundary when drawing inequalities (because the "<" sign does not include the boundary itself, unlike "≤").
- Shade the area below the line because it represents [tex]\( -x + 4y < 12 \)[/tex].
### Here's how the graph should look:
```
y
↑
| /
| /
| /
| /
| /
| /
|/___________> x
```
- The line should intersect the y-axis at 3 (on the y-axis when [tex]\( x = 0 \)[/tex]).
- The x-axis should be intersected at the point when [tex]\( y = 0 \)[/tex], so set [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = \frac{1}{4}x + 3 \implies x = -12 \][/tex]
With the proper graphing tools, your plot will be even more precise.
### Step 1: Rewrite the inequality as an equation
First, we convert the inequality into an equation to find the boundary line:
[tex]\[ -x + 4y = 12 \][/tex]
### Step 2: Solve for [tex]\( y \)[/tex] to get the equation in slope-intercept form
Solving for [tex]\( y \)[/tex]:
[tex]\[ 4y = x + 12 \][/tex]
[tex]\[ y = \frac{1}{4}x + 3 \][/tex]
This tells us that the line has a slope of [tex]\( \frac{1}{4} \)[/tex] and a y-intercept of 3.
### Step 3: Plot the boundary line
We plot the line [tex]\( y = \frac{1}{4}x + 3 \)[/tex]. This line will divide the plane into two regions.
### Step 4: Determine the shading (inequality region)
We need to determine which side of the line to shade for the inequality [tex]\( -x + 4y < 12 \)[/tex].
Choose a test point that is not on the line; the origin [tex]\((0,0)\)[/tex] is often a convenient choice. Substitute [tex]\((0,0)\)[/tex] into the inequality:
[tex]\[ -0 + 4(0) < 12 \][/tex]
[tex]\[ 0 < 12 \][/tex]
This is true, so the region that includes the origin is the solution to the inequality. This means we shade the region below the line.
### Step 5: Draw the graph
1. Plot the line [tex]\( y = \frac{1}{4}x + 3 \)[/tex]:
- The y-intercept is at [tex]\( (0, 3) \)[/tex].
- The slope [tex]\( \frac{1}{4} \)[/tex] means that for every increase of 1 unit in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\( \frac{1}{4} \)[/tex] units. Therefore, you can plot another point by moving 4 units to the right (positive [tex]\( x \)[/tex]) and 1 unit up (positive [tex]\( y \)[/tex]).
- Two points to plot could be [tex]\( (0, 3) \)[/tex] and [tex]\( (4, 4) \)[/tex], then draw the line through these points.
2. Shade the region below the line:
- As determined earlier, the region below this line (including the test point (0,0)) will be shaded to represent [tex]\( -x + 4y < 12 \)[/tex].
### Sketch of the graph:
- Draw the line with points like [tex]\((0,3)\)[/tex] and [tex]\((4,4)\)[/tex].
- Dashed lines usually represent the boundary when drawing inequalities (because the "<" sign does not include the boundary itself, unlike "≤").
- Shade the area below the line because it represents [tex]\( -x + 4y < 12 \)[/tex].
### Here's how the graph should look:
```
y
↑
| /
| /
| /
| /
| /
| /
|/___________> x
```
- The line should intersect the y-axis at 3 (on the y-axis when [tex]\( x = 0 \)[/tex]).
- The x-axis should be intersected at the point when [tex]\( y = 0 \)[/tex], so set [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = \frac{1}{4}x + 3 \implies x = -12 \][/tex]
With the proper graphing tools, your plot will be even more precise.
