Graph the system if inequalities?

Question

Graph the system of inequalities (shade each half plane solution, not only the overall solution.

y <= 3/2 x + 3

2x + y < 10

y > -1

Answer 1

The inequalities are y ≤ (3/2)x + 3 , 2x + y  < 10 and y  > - 1.

Rewrite the inequalities so that solves for y  , That's the slope intercept form and it will make the boundary line easier to graph.

1) Draw the coordinate plane.

The first inequality y ≤ (3/2)x + 3 .

2) Graph the line y = (3/2)x + 3

3) Since the inequality symbol is  ≤  the boundary is included the solution set.

Graph the boundary of the inequalityy ≤ (3/2)x + 3  with solid line.

4) To determine which half plane to be shaded use a test point in either half- plane.

A simple chioce is (0,0). Substitute x  = 0 and y  = 0 in original inequality y ≤ (3/2)x + 3

(0) ≤ (3/2)(0) + 3

0 ≤  3.

The statement is true.

5) Since the statement is true , shade the region contains the point (0,0),Shaded in maroon colour.

The second inequality 2x + y  < 10.

y  < - 2x + 10.

6) Graph the line y  = - 2x + 10.

7) Since the inequality symbol is < the boundary is not included the solution set.

Graph the boundary of the inequality y  < - 2x + 10  with dotted line.

8) To determine which half plane to be shaded use a test point in either half- plane.

A simple chioce is (0,0). Substitute x  = 0 and y  = 0 in original inequality y  < - 2x + 10.

0 < - 2(0) + 10

0 < 10

The statement is true.

9) Since the statement is true , shade the region contains the  point (0,0), shaded in blue colour.

The third inequality y > - 1.

10) Graph the line y  = - 1.

11) Since the inequality symbol is  >  the boundary is not included the solution set.

Graph the boundary of the inequality y > - 1  with dotted line.

12) To determine which half plane to be shaded use a test point in either half- plane.

A simple chioce is (0,0). Substitute x  = 0 and y  = 0 in original inequality y  > - 1.

0 > - 1.

The statement is true.

13) Since the statement is true , shade the region contains the point (0,0), shaded in green colour.

Graph :

The solution of the system is the set of ordered pairs in the intersection of the graph of

y ≤ (3/2)x + 3 , 2x + y  < 10 and y  > - 1. This region is shaded in dark purple color.