) Represent the following inequalities graphically by shading the unwanted region
x≥0, y≥0, x+y≥5, x+y≤10, y≤7, x≤7
draw each of the lines, and shade left or right, above or below, as needed.
To represent the given inequalities graphically and shade the unwanted region, we will follow these steps:
Step 1: Start by graphing the lines corresponding to the inequalities.
x ≥ 0:
This inequality represents all the x-values greater than or equal to zero. It forms a vertical line passing through the y-axis at x = 0.
y ≥ 0:
This inequality represents all the y-values greater than or equal to zero. It forms a horizontal line passing through the x-axis at y = 0.
x + y ≥ 5:
This inequality represents all the points that lie above the line x + y = 5. To graph this line, we find two points on the line by substituting x = 0 and y = 5 into the equation. The points are (0, 5) and (5, 0). We connect these two points with a solid line.
x + y ≤ 10:
This inequality represents all the points that lie below the line x + y = 10. To graph this line, we find two points on the line by substituting x = 0 and y = 10 into the equation. The points are (0, 10) and (10, 0). We connect these two points with a solid line.
y ≤ 7:
This inequality represents all the points that lie below the line y = 7. We draw a horizontal line passing through y = 7.
x ≤ 7:
This inequality represents all the points that lie to the left of the line x = 7. We draw a vertical line passing through x = 7.
Step 2: Shade the unwanted region
Since we need to find the region that satisfies all the given inequalities, we need to shade the region where all the shaded areas overlap.
Shade the region above the line x + y = 5 with a diagonal pattern. This is the area satisfying the inequality x + y ≥ 5.
Shade the region below the line x + y = 10 with horizontal lines. This is the area satisfying the inequality x + y ≤ 10.
Shade the region below the line y = 7 with vertical lines. This is the area satisfying the inequality y ≤ 7.
Shade the region to the left of the line x = 7 with diagonal lines. This is the area satisfying the inequality x ≤ 7.
After shading all the regions, the intersection of all the shaded areas is the region that satisfies all the given inequalities.
Here is a graphical representation of the inequalities:
```
|\
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| \
_________|___\_____
| \
| \
| \
| \
| \
| \
| \
__________|__________\_____
| \
| \
| \
| \
| \
| \
| \
|__________________\________
7 10
```
The shaded region is the triangle formed by the lines x + y = 5, x + y = 10, and y = 7, bounded by the x-axis and y-axis, and x ≤ 7.
Let me know if I can help you with anything else.
To represent the given inequalities graphically and shade the unwanted region, we need to plot the lines corresponding to each inequality and shade the region that satisfies all the given conditions.
Let's consider each inequality one by one and graph them accordingly:
1. x ≥ 0:
This inequality represents all the x-values greater than or equal to 0. It means that the desired region lies to the right of the y-axis, including the y-axis itself. Shading the region to the right side of the y-axis will represent this inequality.
2. y ≥ 0:
This inequality represents all the y-values greater than or equal to 0. It means that the desired region lies above the x-axis, including the x-axis itself. Shading the region above the x-axis will represent this inequality.
3. x + y ≥ 5:
To graph this inequality, we first convert it to an equation: x + y = 5. Plotting this line on the coordinate plane, we get a line passing through points (5, 0) and (0, 5). Since we are interested in values greater than or equal to 5, we need to shade the region above the line.
4. x + y ≤ 10:
To graph this inequality, we first convert it to an equation: x + y = 10. Plotting this line on the coordinate plane, we get a line passing through points (10, 0) and (0, 10). Since we are interested in values less than or equal to 10, we need to shade the region below the line.
5. y ≤ 7:
This inequality represents all the y-values less than or equal to 7. It means that the desired region lies below the line y = 7. Shading the region below the line will represent this inequality.
6. x ≤ 7:
This inequality represents all the x-values less than or equal to 7. It means that the desired region lies to the left of the line x = 7. Shading the region to the left of the line will represent this inequality.
Now, to find the region that satisfies all the given conditions, we need to shade the region that represents the intersection of all the shaded regions from each inequality. The resulting shaded region is the unwanted region.
After plotting and shading all the individual regions described in the above steps, the unwanted region would be the area outside the intersection of all these shaded regions.