Cheyanne Leigh

2021-08-04

Systems of Inequalities Graph the solution set of the system if inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded.

Szeteib

Skilled2021-08-05Added 102 answers

To graph:

The given system of inequality. Also find the coordinates of all vertices, and check whether the solution set is bounded.

Graph:

The given system of the inequalities is,

The corresponding equation of the inequality (1) is,

Since, the inequality

Therefore, the parabola

The corresponding equation of the inequality (2) is,

Since, the inequality

Therefore, the line

Consider the test points (0,4) to check whether the solution satisfies each inequality of the given system.

Substitute 0 for x and 4 for y in the inequality

The point (0,4) is inside the parabola

Substitute 0 for x and 4 for y in the inequality

The points (0,4) is below the line

Therefore, the test point(0,4) satisfies each inequality of the given system.

The solution set of the given system of inequalities is the intersection of the solutions of each of the given inequality.

Therefore, the solution set is shown as shaded region.

The vertices occur at the points of intersection of the corresponding equation of the given system of inequalities.

It is observed from Figure 1 that the parabola

Substitute

Further solve the above equation for the value of x.

Therefore, the x-coordinate of vertex are -3 and 2.

Substitute -3 for x in equation (3).

Substitute 2 for x in exuation (3).

Therefore, the y-coordinate of vertex are 0 and 5.

Therefore, the vertices of the shaded region are (-3,0) and (2,5).

It is observed from Figure 1 that the shaded region is enclosed by the boundary lines of the given system of inequalities.

Therefore, the shaded region is bounded.

Interpretation:

The solution set of the given system of inequality lies in 1 and 2 quadrant as shown in Figure 1.

Conclusion:

Thus, the vertices of the given system of inequalities are (-3,0) and (2,5), the solution set is bounded.

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