y ≥ −3x + 1
y ≤ 1 over 2x + 3
To graph this system of inequalities, we will first graph the boundary lines for each inequality and then shade the regions that satisfy both inequalities.
1. y ≥ −3x + 1
To graph this line, we will first find its x and y-intercepts.
When y=0, -3x+1=0, so x=1/3. When x=0, y=1. Plotting these on a coordinate grid and drawing the line through these two points, we get:
graph{y>=-3x+1 [-6, 6, -3, 5]}
2. y ≤ 1 over 2x + 3
To graph this line, we will first find its x and y-intercepts.
When y=0, 1/2x+3=0, so x=-6. When x=0, y=1/3. Plotting these on a coordinate grid and drawing the line through these two points, we get:
graph{y<=1/(2x+3) [-6, 6, -4, 4]}
3. Shading the region that satisfies both inequalities
To shade the region that satisfies both inequalities, we need to determine which side of each line to shade. Since y is greater than or equal to -3x+1 in the first inequality, we will shade the region above the line. Since y is less than or equal to 1/(2x+3) in the second inequality, we will shade the region below the line. The overlapping region that satisfies both inequalities is the shaded region:
graph{y>=-3x+1 and y<=1/(2x+3) [-6, 6, -4, 4]}
Therefore the solution to the system of inequalities is the shaded region, which satisfies both conditions.
To solve this system of inequalities, we need to find the region on the coordinate plane that satisfies both inequalities.
Let's start with the first inequality, y ≥ -3x + 1:
Step 1: Plot the line y = -3x + 1.
To do this, we need to find two points on the line. We can start by setting x to 0 and solving for y. When x = 0, y = -3(0) + 1 = 1. So, we have the point (0, 1).
Next, we can set y to 0 and solve for x. When y = 0, 0 = -3x + 1. Solving for x, we get x = 1/3. So, we have the point (1/3, 0).
Step 2: Graph the line y = -3x + 1.
Draw a line through the two points (0, 1) and (1/3, 0).
Step 3: Shade the region above the graph of y = -3x + 1.
Since the inequality is y ≥ -3x + 1, we need to shade the region above the graph of the line.
Now let's move on to the second inequality, y ≤ 1/2x + 3:
Step 4: Plot the line y = 1/2x + 3.
Again, we need to find two points on the line. When x = 0, y = (1/2)(0) + 3 = 3. So, we have the point (0, 3).
When y = 0, 0 = (1/2)x + 3. Solving for x, we get x = -6. So, we have the point (-6, 0).
Step 5: Graph the line y = 1/2x + 3.
Draw a line through the two points (0, 3) and (-6, 0).
Step 6: Shade the region below the graph of y = 1/2x + 3.
Since the inequality is y ≤ 1/2x + 3, we need to shade the region below the graph of the line.
Step 7: Determine the region that satisfies both inequalities.
To find the region that satisfies both inequalities, we need to find the overlapping shaded regions from Step 3 and Step 6.
In this case, the overlapping shaded region represents the solution to the system of inequalities y ≥ -3x + 1 and y ≤ 1/2x + 3.
Note: The overlapping region may be a single point, a line segment, or a larger region depending on the specific inequalities given.