Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.
y, is greater than or equal to, start fraction, 3, divided by, 2, end fraction, x, minus, 8
y≥
2
3
x−8
y, is less than, minus, x, minus, 3
y<−x−3
To graphically solve the system of inequalities, we will plot the lines corresponding to each inequality and shade the region that satisfies both inequalities.
First, let's graph the line y = (3/2)x - 8:
To do this, we can find the x- and y-intercepts:
When x = 0, y = (3/2)(0) - 8 = -8.
When y = 0, 0 = (3/2)x - 8, so x = (8)(2/3) = 16/3 ≈ 5.33.
Plotting the x-intercept (16/3, 0) and the y-intercept (0, -8) gives the line:
{{{graph(300,300,-5,10,-15,5,(3/2)x - 8)}}}
Next, let's graph the line y = -x - 3:
To do this, we can find the x- and y-intercepts:
When x = 0, y = -(0) - 3 = -3.
When y = 0, 0 = -x - 3, so x = -3.
Plotting the x-intercept (-3, 0) and the y-intercept (0, -3) gives the line:
{{{graph(300,300,-5,10,-15,5,(3/2)x - 8, -x - 3)}}}
Now we need to find the shaded region that satisfies both inequalities.
Since y is greater than or equal to (3/2)x - 8, we will shade the region above or on the line y = (3/2)x - 8.
Also, since y is less than -x - 3, we will shade the region below the line y = -x - 3.
The shaded region that satisfies both inequalities is the area below the line y = -x - 3 and above or on the line y = (3/2)x - 8.
The coordinates of a point in the solution set can be any point within this shaded region.
For example, one possible point in the solution set could be (-1, -4).
Please note that there are infinitely many points in the solution set.