Graph the inequality and show all your work y<-2/3x+4 Please I really need help!?

Question

Graph the system of inequalities. Clearly label the boundary lines. Then use your graph to identify any two points that are solutions to the system. Show and explain your work. 

f(x)={y≤1/2x+2 
x+y>-3 

Answer 1

(1).

The inequality is y < - (2/3)x + 4.

The graph of the inequality y < - (2/3)x + 4 is the shaded region, so every point in the shaded region satisfies the inequality.

The graph of the equation y = - (2/3)x + 4 is the boundary of the region. Since the inequality <, the boundary is drawn as a dashed line to show that points on the line does satisfy the inequality.

 

Graph the boundary line by using the x - intercept and y - intercept of the line.

To find the y - intercept, substitute the value of x = 0 in the original equation.

y = - (2/3)(0) + 4 ⇒ y = 4.

The y - intercept is 4, so the graph intersects the y - axis at (0, 4).

 

To find the x - intercept, substitute the value of y = 0 in the original equation.

(0) = - (2/3)x + 4 ⇒ x = 6.

The x - intercept is 6, so the graph intersects the x - axis at (6, 0).

 

To draw inequality y < - (2/3)x + 4 follow the steps.

1.  Draw a coordinate plane.

2.  Plot the points and draw a line through these points.

3.  To determine which half-plane to be shaded use a test point in either half-plane. A simple choice is (0, 0).

Substitute the value of (x, y) = (0, 0) in the original inequality.

(0) < - (2/3)(0) + 4 ⇒ 0 < 4.

4.  Since the above statement is true, shade the region that contains point (0, 0).

The inequality y < - (2/3)x + 4 graph is

 

Answer 2

(2).

The system of inequalities are y ≤ (1/2)x + 2 and x + y > -3.

Therefore system of equalities are y = (1/2)x + 2 and x + y = -3.

For ease of graphing, write each equation in slope - intercept form.

Equation 2 : x + y = -3 ⇒ y = - x - 3.

1. Draw a coordinate plane.

2. Since the symbol of inequality y ≤ (1/2)x + 2 is <, the boundary is included in the solution set. Graph the boundary of the equality y = (1/2)x + 2 with solid line.

3. Since the symbol of inequality x + y > -3 is >, the boundary is not included in the solution set. Graph the boundary of the equality x + y = -3 with dashed line.

5. The solution of the system is the set of ordered pairs in the intersection of the graph of y ≤ (1/2)x + 2 and x + y > -3. This region is shaded in red color.

6. CHECK :

Choose a point in the red color region, such as (2, 2). To check this solution, substitute 2 for x and 2 for y into each inequality.

Choose a point in the red color region, such as (4, -4). To check this solution, substitute 4 for x and -4 for y into each inequality.