Graph of the System of Inequalities

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Graph the solution of the system of inequalities and find the coordinates of all vertices for

a. x + y ≤ 4
          y ≥ x

b. 2x + 3y > 12
     3x - y < 21

Solution:

a. For x + y ≤ 4, we need to rewrite the given equation into slope-intercept form as follows

Since the sign of inequality is less than or equal to, then all points along the line are included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 ≤ 4 which is correct and that point is included in the solution.  

For y ≥ x, the given equation is already written in slope-intercept form. 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. Therefore, the graph of a pair of linear inequalities is

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The vertex of the graph is (2, 2) which is the intersection of two lines. It is also included in the solution.

b. For 2x + 3y > 12, we need to rewrite the given equation into slope-intercept form as follows

Since the sign of inequality is greater 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 > 12 which is not correct and that point is not included in the solution.  

For 3x - y < 21, 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 < 21 which is correct and that point is  included in the solution. Therefore, the graph of a pair of linear inequalities is

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The vertex of the graph which is the intersection of two lines  is not included in the solution.

Graphs of Quadratic Inequalities

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Graph the following inequalities:

a. y > x² + 1
b. x² + y² ≥ 4

Solution:

a. For y > x² + 1, the given equation is a parabola that concave upward whose vertex is V(0, -1). Since the sign of inequality is greater than, then all points along the curve are not included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 > 1 which is not correct and that point is not included in the solution. Therefore, the graph of the given equation is

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b. For x² + y² ≥ 4, the given equation is a circle whose center is C(0, 0) and its radius is r = 2. 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 ≥ 4 which is not correct and that point is not included in the solution. Therefore, the graph of the given equation is

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Graphs of Linear Inequalities

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Graph the following inequalities:

a. y > -3
b. x ≤ 2
c. 3x + 4y + 12 > 0
d. 2x - y ≤ 8

Solution:

a. For y > -3, the given line is a horizontal line. Since the sign of inequality is greater than, then all points along the line are not included in the solution. However, all points above the given line are included in the solution. Here's the graph of the given line

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b. For x ≤ 2, the given line is a vertical line. Since the sign of inequality is less than or equal to, then all points along the line are included in the solution. However, all points at the left of the given line are included in the solution. Here's the graph of the given line

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c. For 3x + 4y + 12 > 0, we need to rewrite the given equation into slope-intercept form as follows
 

 

 

Since the sign of inequality is greater 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 12 > 0 which is correct and that point is included in the solution. Therefore, the graph of the given equation is
 

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d. For 2x - y ≤ 8, 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 or equal to, then all points along the line 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. Therefore, the graph of the given equation is

Photo by Math Principles in Everyday Life