Explanation
Problem #2
We must find the solution to the following system of inequalities:
[tex]\begin{gathered} 3x-2y\leq4, \\ x+3y\leq6. \end{gathered}[/tex]
(1) We solve for y the first inequality:
[tex]-2y\leq4-3x.[/tex]
Now, we multiply both sides of the inequality by (-1), this changes the signs on both sides and inverts the inequality symbol:
[tex]\begin{gathered} 2y\ge-4+3x, \\ y\ge\frac{3}{2}x-2. \end{gathered}[/tex]
The solution to this inequality is the set of all the points (x, y) over the line:
[tex]y=\frac{3}{2}x-2.[/tex]
This line has:
• slope m = 3/2,
,
• y-intercept b = -2.
(2) We solve for y the second inequality:
[tex]\begin{gathered} x+3y\leq6, \\ 3y\leq6-x, \\ y\leq-\frac{1}{3}x+2. \end{gathered}[/tex]
The solution to this inequality is the set of all the points (x, y) below the line:
[tex]y=-\frac{1}{3}x+2.[/tex]
This line has:
• slope m = -1/3,
,
• y-intercept b = 2.
(3) Plotting the lines of points (1) and (2), and painting the region:
• over the line from point (1),
,
• and below the line from point (2),
we get the following graph:
Answer
The points that satisfy both inequalities are given by the intersection of the blue and red regions:
