How do you graph the inequality #–3x – 4y

Question

How do you graph the inequality #–3x – 4y<=12#?

Ethan Adkins · Accepted Answer

To graph this kind of inequality, the best thing is to manipulate them and obtain a relation of the form #y\le f(x)# (or #y \ge f(x)#). In fact, we know that #y=f(x)# is exactly the graph of #f#, and so #y\le f(x)# (or #y \ge f(x)#) represents all the portion of the plan below (or above) the graph of #f#. Let's do those manipulations: starting from 
#-3x-4y\le 12#,
adding #4y# at both sides we get
#-3x \le 4y+12#.
Subtracting 12 at both sides, we have
#-3x-12\le 4y#.
Dividing both sides by #4# we finally have
#-3/4 x - 3 \le y#
which can of course be read as
#y \ge -3/4 x - 3# We thus have #f(x)=-3/4 x - 3#, which is a line and so it's very easy to graph. Once graphed, you need to consider all the portion of plan above the line to solve your inequality.  Here's the graph: graph{-3x-4y \le 12 [-10, 10, -5, 5]}

Abigail Allen · Answer

undefined To graph the inequality -3x - 4y ≤ 12:

1. Begin by graphing the corresponding equation, -3x - 4y = 12:
   - Plot the x-intercept by setting y = 0 and solving for x: -3x = 12, x = -4. Plot the point (-4, 0).
   - Plot the y-intercept by setting x = 0 and solving for y: -4y = 12, y = -3. Plot the point (0, -3).
   - Draw a straight line passing through these two points.

2. Since the inequality includes ≤ (less than or equal to), the region below or on the line is shaded.

3. Choose a test point not on the line, like (0, 0), and substitute its coordinates into the original inequality (-3(0) - 4(0) ≤ 12). If the inequality holds true, shade the region containing the test point. Otherwise, shade the opposite region.

4. Extend the shaded region from step 3 to cover the entire area below or on the line.

5. Label the shaded region to indicate that it represents the solution set for the inequality.

How do you graph the inequality #–3x – 4y&lt;=12#?

How do you graph the inequality #–3x – 4y<=12#?