Hello!
The graph is attached.
Why?
To find the graph that shows the solution to the system of linear inequalities below, we need to find the axis intercepts and consider the given conditions for both inequalities.
So, solving we have:
First inequality: Blue line and area.
[tex]y\leq -3x+2[/tex]
Finding the x-axis intercept, making "y" equal to 0.
[tex]y=-3x+2[/tex]
[tex]0=-3x+2\\\\3x=2\\\\x=\frac{2}{3}=0.66[/tex]
So, the x-axis intercept is located at the point (0.66,0)
Finding the y-axis intercept, making "x" equal to 0.
[tex]y=-3x+2[/tex]
[tex]y=-3*0+2=2[/tex]
So, the y-axis intercept is located at the point (0,2)
Also, we can see that since the coefficient of the linear term is negative (-3) the line is decreasing (line with negative slope).
Now, that we already know the axis intercepts, we need to remember that the inequality is given by the following expression:
[tex]y\leq -3x+2[/tex]
The less or equal inequality symbol means that the solution region of the inequality is under the function.
Second inequality: Red line and area.
[tex]y>2x-3[/tex]
Finding the x-axis intercept, making "y" equal to 0.
[tex]y=2x-3+2[/tex]
[tex]0=2x-3\\\\2x=3\\\\x=\frac{3}{2}=1.5[/tex]
So, the x-axis intercept is located at the point (1.5,0)
Finding the y-axis intercept, making "x" equal to 0.
[tex]y=2x-3[/tex]
[tex]y=2*0-3=-3[/tex]
So, the y-axis intercept is located at the point (0,-3).
Also, we can see that since the coefficient of the linear term is positive (-3) the line is increasing (line with positive slope).
Now, that we already know the axis intercepts, we need to remember that the inequality is given by the following expression:
[tex]y>2x-3[/tex]
The greater inequality symbol means that the solution region of the inequality is above the function.
Hence, since we have a system of linear inequalities, the solution to the whole system is located between the solution region of the given inequalities.
The graph is attached, the solution to the system of linear inequalities is represented by the shaded area (light purple area).
Have a nice day!
