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Graph the solution of the system of inequalities and find the coordinates of all vertices for
a. x² + y² ≤ 8
x ≥ 2
y ≥ 0
b. x² - y ≥ 0
x + y < 6
x - y < 6
Solution:
a. For x² + y² ≤ 8, the given equation is a circle whose center is C(0, 0) and r = √8 = 2√2. Since the sign of inequality is less than or equal to, then all points along the curve are included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 ≤ 8 which is correct and that point is included in the solution.
For x ≥ 2, the given line is a vertical line in which all points along the line are included in the solution. However, all points at the right of the given line are included in the solution.
For y ≥ 0, the given line is a horizontal line in which all points along the line are included in the solution. However, all points above of the given line are included in the solution. Therefore, the graph of a set of inequalities is
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| Photo by Math Principles in Everyday Life |
The vertices of the graph are (2, 2), (2, 0) and (2√2, 0) that are located at the intersection of the three shaded regions bounded by two lines and a circle. They are also included in the solution.
b. For x² - y ≥ 0, the given equation is a parabola that concave upward whose vertex is V(0, 0). Since the sign of inequality is greater than or equal to, then all points along the curve are included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 ≥ 0 which is correct and that point is included in the solution.
For x + y < 6, we need to rewrite the given equation into slope-intercept form as follows
Since the sign of inequality is less than, then all points along the line are not included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 < 6 which is correct and that point is included in the solution.
For x - y < 6, we need to rewrite the given equation into slope-intercept form as follows
If you divide both sides of the equation by a negative number, then the sign of inequality will be reversed. Since the sign of inequality is less than, then all points along the line are not included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 < 6 which is correct and that point is included in the solution. Therefore, the graph of a set of inequalities is
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| Photo by Math Principles in Everyday Life |
The vertices of the graph are (-3, 9) and (2, 4) that are located at the intersection of the three shaded regions bounded by two lines and a parabola. They are also included in the solution.