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, minus, 3, x, minus, 3
y>−3x−3
y, is greater than, x, minus, 7
y>x−7
Here is the graph of the system of inequalities:
First, let's graph the line y = -3x - 3.
To do this, we can start by graphing the line y = -3x, which has a y-intercept of 0 and a slope of -3. Then, we can shift the line downward by 3 units by moving each point on the line down 3 units.
Next, let's graph the line y = x - 7.
To do this, we can start by graphing the line y = x, which has a y-intercept of 0 and a slope of 1. Then, we can shift the line downward by 7 units by moving each point on the line down 7 units.
Now, we need to shade the region above the line y = -3x - 3 and above the line y = x - 7.
The shaded region represents the solution set of the system of inequalities.
A point in the solution set is (1, -2) since it's located in the shaded region.
Here is the graph:
|
x |
-10 | . . . . . .
| /
-9 | . . . . .
| /
-8 | . . . .
| /
-7 | . . .
| /
-6 | . .
|/
-5 | .
--------------
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
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 less than, minus, start fraction, 3, divided by, 2, end fraction, x, minus, 7
y<−
2
3
x−7
y, is less than, one half, x, plus, 1
y<
2
1
x+1
Here is the graph of the system of inequalities:
First, let's graph the line y = (3/2)x - 7.
To do this, we can plot two points on the line. For example, when x = 0, y = -7, and when x = 2, y = -4. We can then connect these two points to graph the line.
Next, let's graph the line y = (1/2)x + 1.
To do this, we can plot two points on the line. For example, when x = 0, y = 1, and when x = 2, y = 2. We can then connect these two points to graph the line.
Now, we need to shade the region below the line y = (3/2)x - 7 and below the line y = (1/2)x + 1.
The shaded region represents the solution set of the system of inequalities.
A point in the solution set is (4, -2) since it's located in the shaded region.
Here is the graph:
|
x |
-10 |
|
-8 | .
| |
-6 | . | . . .
| | |
-4 | . | |
| | | |
-2 | | | |
| | | |
0 |_____________
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
To solve the system of inequalities graphically, we will first draw the graph of each individual inequality and then find their intersection region.
1. Start by drawing the graph of the inequality y > -3x - 3:
- Note that this is an inequality in slope-intercept form (y = mx + b), where m is the slope (-3 in this case) and b is the y-intercept (-3 in this case).
- To draw the graph, you can plot the y-intercept at -3 on the y-axis, then use the slope to find one or two more points and connect them to form a line.
- Since it is a strict inequality (greater than), the line should be dashed to indicate that points on the line are not included in the solution set.
2. Next, draw the graph of the inequality y > x - 7:
- Again, this is in slope-intercept form (y = mx + b), where m is the slope (1 in this case) and b is the y-intercept (-7 in this case).
- Plot the y-intercept at -7 on the y-axis, then use the slope to find additional points and connect them to form a line.
- This is also a strict inequality, so the line should be dashed.
3. The solution to the system of inequalities is the intersection region of the two graphs.
- Shade the region where both inequalities are satisfied, which is above both lines (since y is greater than).
- The point(s) in this shaded region represent the solution points for the system of inequalities.
4. Identify a point in the solution set:
- Choose any point within the shaded region.
- Read the coordinates of the chosen point (x, y) to state a point in the solution set.
Note: Since I'm an AI text-based model, I can't draw the graph for you. However, you can use graphing tools or software like Desmos or GeoGebra to input the inequalities and visualize the solution graph.