How to plot the following in coordinate system

[goelmohit2002]

Hi All,

Can someone please tell how to plot the following equation area in coordinate system and what is the reasoning for the same ?

2x + 3y <= 6

I was able to draw the line that represent 2x + 3y = 6. But how to draw the area that represents 2x + 3y <= 6.

Thanks
Mohit

[rohit3249]

All the area that is below the line 2x+3y-6=0 will be covered.

[goelmohit2002]

All the area that is below the line 2x+3y-6=0 will be covered.

thanks. Can you please tell why ?

[RonPurewal]

All the area that is below the line 2x+3y-6=0 will be covered.

thanks. Can you please tell why ?

there are 2 ways to do this:

(1) DON'T rearrange to mx + b; TEST A POINT
this is the way you've done it.

if you DON'T rearrange the equation to the form y = mx + b, but instead leave it in the form in which you found it - as you did here - then you probably won't be able to tell from just looking at the equation which side to shade.

in this case, just test any point that's not on the boundary line. (in this case, this means any point for which 2x + 3y is not equal to 6.)
if the point WORKS in the original inequality, then shade the side CONTAINING the point.
if the point DOES NOT WORK in the originla inequality, then shade the side that DOES NOT CONTAIN the point.

the easiest such point here, by far, is (0, 0). (this is always the easiest point to check, unless it's on the line in question.)
if you plug (0, 0) into 2x + 3y < 6, then the inequality is true. therefore, you should shade the side containing (0, 0).

(2) REARRANGE TO mx + b

if you take the time to rearrange into mx + b form:
if it's y > mx + b or y > mx + b, then you shade ABOVE the line.
if it's y < mx + b or y < mx + b, then you shade BELOW the line.

here, if you solve for mx + b form, you'll get y < (-2/3)x + 2. since the sign is "<", you shade BELOW the line.

[RonPurewal]

@ mohit

by the way, two things.

(1) you don't have to write "<=" and ">=", which are hard to read, for inequality signs. instead, you can make "<" into "<", or ">" into ">", by just underlining them.

(2) this is a routine algebra topic. that doesn't mean that you shouldn't post it here, but you can get INSTANT answers by just entering a google search of what you're looking for.
in this case, i typed "shading linear inequality" into google and got 5-6 first page hits that explain EXACTLY what you're looking for.

[goelmohit2002]

(2) REARRANGE TO mx + b

if you take the time to rearrange into mx + b form:
if it's y > mx + b or y > mx + b, then you shade ABOVE the line.
if it's y < mx + b or y < mx + b, then you shade BELOW the line.

here, if you solve for mx + b form, you'll get y < (-2/3)x + 2. since the sign is "<", you shade BELOW the line.

Thanks a lot Ron !!!

Can you please tell is this the case with both +ve and -ve slope lines ? i.e. does both ascending and descending lines follow this ?

[Ben Ku]

What Ron stated about which side of the line (written in y=mx+b) is true regardless of whether the slope is positive or negative. However, if you are unsure, it's safest to try a test point.