Graph Linear Inequalities

with 9 comments


How to graph a linear inequality. When to use a dotted, or solid line, and which side to shade on.




Welcome to Your Tutor Online video lessons. Today will we learn how to graph linear inequalities. A linear inequality is an equation for a line. That means it will have two variables and it has a less than, greater than, less than or equal to, or great than or equal to sign instead of the equals sign. Here we have an example 2x minus 3y is less than 6.


For the most part we are just going to pretend that this less than sign is an equal sign to solve the equation into slope intercept form. The only exception is if we multiple or divide by a negative number. When that happens, flip the sign so that it points in the other direction.


To understand why we need to flip the sign lets look at this very simply example. 3 is less than 5 which is true. Now lets multiply everything by negative 1. In a normal equation this would be fine, as long as you do something to one side that you do to the other you should maintain the equation. But here we’ll get negative 3 is less than negative 5 which is not true. Negative 5 is less than negative 3. So to account for that, instead we are going to flip the sign whenever we multiply or divide by a negative. Negative 3 is greater than negative 5, which is true.


Now let’s go back to our example and put it into slope intercept form. We are going to subtract 2x from both sides. That leaves us with negative 3y is less than negative 2x plus 6. Now we divide by negative 3 and we are left with y on this side. We need to flip our sign since we are dividing by a negative number. So now it’s greater than. Negative 2 divided by negative 3 is positve two-thirds-x and 6 divided by negative 3 is negative 2.


Now that the equation is in slope intercept form, graph it as you normally would. I’m going to start at the y axis, negative 2 for the intercept, is here. Now we are going to follow our slope – up two, over three. We are going to do that one more time: up two over three. Now, before we connect the dots we need to go back and look at the equation. Graphing here is slightly different than a normal line. If the inequality has a less than or greater than sign, we will use the dashed line. If it is less than or eqaul to, or greater than or equal to we will use a solid line. If “equal” is in the name of the symbol then use a normal line. For our example, it’s greater than so we use the dashed line. We are going to use a dashed line because the points on this line are not included in the equation.


There’s one extra thing we need to do when graphing inequalities. We have to shade on one side of the line. To figure out which one, pick any point that is not on the line. I always use the point zero, zero when I can because it is the easiest to work with. Remember that in ordered pairs the first number is x and the second number is y. We are going to take those two numbers and plug them into our inequality.


Zero is greater than zero times two-thirds x is zero; negative two. Zero is greater than negative two. If that statement is true then we will shade on the side of the line that contains the point we tested for. If its false then we will just shade on the opposite side.


So, zero is greater than negative two is true, so we shade on this side of the dotted line. And thats all there is to graphing inequalities.


I hope you found this lesson useful. If you have any questions leave a comment on the blog at www.YourTutorOnline.com If you have any lessons suggestions send an email to podcast[at]yourtutoronline.com Thanks for watching, class dismissed.

Written by Joel

September 8th, 2008 at 11:00 am

Posted in Lessons

9 Responses to 'Graph Linear Inequalities'

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  1. thanks


    1 Oct 08 at 1:19 pm

  2. thanks that helped me alot


    5 Oct 08 at 11:20 pm

  3. I’m glad you found that useful. Thats what I’m here for.

    Thanks for leaving your comments, I really appreciate it.


    6 Oct 08 at 6:46 am

  4. this was really helpful. thanks a bunch.


    6 Nov 08 at 5:06 pm

  5. I’m glad you found it helpful. Thanks for telling me you liked it!


    6 Nov 08 at 5:17 pm

  6. hey i don’t know this one?
    can you help me out?


    11 Feb 09 at 10:04 pm

  7. Well, first that is not an inequality (which have < or >).

    First change it into slope intercept form (y = mx +b)
    12y = 3x + 12
    y = (1/4)x + 1

    Graph the y-int at (0,1) and then go up 1 and over 4.

    See my other video here:


    12 Feb 09 at 10:37 am

  8. Thanks so much! You saved me for a math test! :)


    13 Dec 10 at 2:18 am

  9. Thank you so much! I have a midterm tomorrow and I had no idea what this was, I keep telling my teacher I don’t understand, and she keeps telling me the same thing!! This really saved my life!!! YourTutuorOnline is know my site 4 studying math !!


    6 May 11 at 6:37 am

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