Solve the following system of linear inequalities by graphing. 3x – y < 2
x + y > 2
we cannot draw graphs for you here. I will instead describe how to do this problem.
Draw the lines y = 3x 2 and y = x +2 on a graph. The yintercepts are 2 and =2 respectively, and the lines slope in opposite directions. The wedgedshaped solution region lies above both of these lines, since y>3x2 and y>x+2
Y<2
To solve the system of linear inequalities by graphing, we will first graph each equation separately and then identify the region of the graph where the solutions to both inequalities overlap.
First, let's graph the equation 3x  y < 2:
To graph this equation, we can start by finding two points that satisfy the equation by letting x = 0 and y = 0.
When x = 0, the equation becomes:
3(0)  y < 2, which simplifies to y < 2.
If we multiply the inequality by 1 (or divide by 1), we need to reverse the inequality sign, so we get:
y > 2
When y = 0, the equation becomes:
3x  0 < 2, which simplifies to 3x < 2.
If we divide both sides by 3, we get:
x < 2/3.
Now, let's plot these two points on a graph and draw a dashed line connecting them:
```
y axis
^



0*
 
  ==> 3x  y < 2
  (dashed line)
2______________________* (2/3, 2)

______________________
0 2/3 x axis
```
Next, let's graph the equation x + y > 2:
To graph this equation, we can again find two points that satisfy the equation by letting x = 0 and y = 0.
When x = 0, the equation becomes:
0 + y > 2, which simplifies to y > 2.
When y = 0, the equation becomes:
x + 0 > 2, which simplifies to x > 2.
Plot these two points on the graph and draw a dashed line connecting them:
```
y axis
^



2______________________*
  ==> x + y > 2
  (dashed line)
 
 
 
__________________*
0 2 x axis
```
Now, we need to shade the region that satisfies both inequalities. The solution region will be where both shaded areas overlap:
```
y axis
^


 
 / / ==> x + y > 2
/____/
 ______
 / / ==> 3x  y < 2
/_____


______________________
0 2/3 x axis
```
From the graph, we can see that the region above the line x + y > 2 and below the line 3x  y < 2 is the solution to the system of inequalities.
To solve the system of linear inequalities, we need to graph each inequality separately and then find the area where the graphs overlap. This overlapping area represents the solution to the system.
Let's start by graphing the first inequality, 3x  y < 2:
Step 1: Rewrite the inequality in the slopeintercept form, which is y = mx + b.
3x  y < 2 (subtract 3x from both sides)
y < 3x + 2 (divide by 1 to make y positive)
y > 3x  2
The inequality y > 3x  2 can be graphed by drawing a dashed line with a slope of 3/1 (rise over run) and a yintercept of 2. Since y is greater than, not equal to, the line, we will make the line dotted.
Step 2: Choose a few test points not on the line to determine which side of the line should be shaded. For example:
Let's choose x = 0:
y > 3(0)  2
y > 2
One point on the side y is greater than 2 is (0, 0).
Step 3: Shade the portion of the graph where y is greater than the line. This can be done by shading above the line.
Now, let's graph the second inequality, x + y > 2:
Step 1: Rewrite the inequality in the slopeintercept form, which is y = mx + b.
x + y > 2 (subtract x from both sides)
y > x + 2
The inequality y > x + 2 can be graphed by drawing a dashed line with a slope of 1 (negative reciprocal of 1) and a yintercept of 2. Again, since y is greater than, not equal to, the line, we will make the line dotted.
Step 2: Choose a few test points not on the line to determine which side of the line should be shaded. For example:
Let's choose x = 0:
y > (0) + 2
y > 2
One point on the side y is greater than 2 is (0, 3).
Step 3: Shade the portion of the graph where y is greater than the line. This can be done by shading above the line.
Now, we have graphed both inequalities. To find the solution to the system, we look for the overlapping shaded region. This region represents the values of x and y that satisfy both inequalities.
In this case, the overlapping region is the area above the line y = 3x  2 and above the line y = x + 2.
Let me know if there is anything further I can assist you with.