Question
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, x, plus, 1
y<−x+1
y, is less than, one fifth, x, plus, 7
y<
5
1
x+7
y, is less than, minus, x, plus, 1
y<−x+1
y, is less than, one fifth, x, plus, 7
y<
5
1
x+7
Answers
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GPT-4o mini
To solve the system of inequalities graphically, we'll consider each inequality and plot them on a set of axes.
1. **First Inequality**: \( y < -x + 1 \)
- The line \( y = -x + 1 \) has a y-intercept of 1 and a slope of -1.
- This means it goes down one unit for each unit it goes to the right.
- To plot the line, we can find two points:
- When \( x = 0 \), \( y = 1 \) → point (0, 1)
- When \( x = 1 \), \( y = 0 \) → point (1, 0)
- The area below this line will represent \( y < -x + 1 \).
2. **Second Inequality**: \( y < \frac{1}{5}x + 7 \)
- The line \( y = \frac{1}{5}x + 7 \) has a y-intercept of 7 and a slope of \( \frac{1}{5} \).
- This means it goes up one unit for every five units to the right.
- To plot the line, we can find two points:
- When \( x = 0 \), \( y = 7 \) → point (0, 7)
- When \( x = 5 \), \( y = 8 \) → point (5, 8)
- The area below this line will represent \( y < \frac{1}{5}x + 7 \).
### Graphing the Inequalities
1. Draw the axes (x and y).
2. Plot the line for \( y = -x + 1 \) using the points (0, 1) and (1, 0). Shade the area below this line.
3. Plot the line for \( y = \frac{1}{5}x + 7 \) using the points (0, 7) and (5, 8). Shade the area below this line.
### Finding the Solution Set
The solution set is the region where the shaded areas of both inequalities overlap. This region represents the values of \( x \) and \( y \) that satisfy both inequalities.
### Example Point in the Solution Set
To find a point in the solution set, we can look for coordinates that lie in the overlapping shaded area. For example, the point (0, 0) can be checked:
- For the first inequality:
- \( 0 < -0 + 1 \) (True)
- For the second inequality:
- \( 0 < \frac{1}{5}(0) + 7 \) (True)
Thus, the point (0, 0) is in the solution set.
### Conclusion
The coordinates of a point in the solution set are: **(0, 0)**.
1. **First Inequality**: \( y < -x + 1 \)
- The line \( y = -x + 1 \) has a y-intercept of 1 and a slope of -1.
- This means it goes down one unit for each unit it goes to the right.
- To plot the line, we can find two points:
- When \( x = 0 \), \( y = 1 \) → point (0, 1)
- When \( x = 1 \), \( y = 0 \) → point (1, 0)
- The area below this line will represent \( y < -x + 1 \).
2. **Second Inequality**: \( y < \frac{1}{5}x + 7 \)
- The line \( y = \frac{1}{5}x + 7 \) has a y-intercept of 7 and a slope of \( \frac{1}{5} \).
- This means it goes up one unit for every five units to the right.
- To plot the line, we can find two points:
- When \( x = 0 \), \( y = 7 \) → point (0, 7)
- When \( x = 5 \), \( y = 8 \) → point (5, 8)
- The area below this line will represent \( y < \frac{1}{5}x + 7 \).
### Graphing the Inequalities
1. Draw the axes (x and y).
2. Plot the line for \( y = -x + 1 \) using the points (0, 1) and (1, 0). Shade the area below this line.
3. Plot the line for \( y = \frac{1}{5}x + 7 \) using the points (0, 7) and (5, 8). Shade the area below this line.
### Finding the Solution Set
The solution set is the region where the shaded areas of both inequalities overlap. This region represents the values of \( x \) and \( y \) that satisfy both inequalities.
### Example Point in the Solution Set
To find a point in the solution set, we can look for coordinates that lie in the overlapping shaded area. For example, the point (0, 0) can be checked:
- For the first inequality:
- \( 0 < -0 + 1 \) (True)
- For the second inequality:
- \( 0 < \frac{1}{5}(0) + 7 \) (True)
Thus, the point (0, 0) is in the solution set.
### Conclusion
The coordinates of a point in the solution set are: **(0, 0)**.
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