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# Intro to graphing two-variable inequalities

Learn how to graph two-variable linear inequalities like y≤4x+3. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• How to I solve compound inequalities?
• Is there any way to find the shaded side easier.
• Draw a little man ⛷ on each line as if it were the side of a mountain. ⛰
The side that's above ground is the greater than side. `≥ or >`
The side that's below ground is the less than side. `≤ or <`

You can also try ✈️ airplane arms and align your own arms with each line.
The side above your shoulders is the greater than side. `≥ or >`
The side below your shoulders is the less than side. `≤ or <`
• Why do you have to put an equal sign in place of the greater and less than signs? And are you supposed to divide or multiply when you have an equation like this -3x-y <-1 ( there is suppose to be a line underneath the less than sign) ?
• The line underneath the greater than or less than sign means less than or equal to and greater than or equal to. Sal did this to show you what this means.
For your second question, you need to divide so you get an x on one side of the equation. For example, if y = 3, than the equation would be -3x-3<=-1. You would then subtract 3 from both sides of the equation to get -3x<=-4. Then, divide both sides by 3 to isolate the x on one side. Since you are dividing by a negative number, reverse the less than or equal to to a GREATER THAN or equal to sign.
(1 vote)
• How do I write the slope if the line goes straight up (is vertical) and how do I write it's equation if the y-intercept is not given but I have an x-intercept?
• If the line goes straight up, then the line's equation is in the form x = ? because only the y value changes, the x value never changes. The x intercept is all you need to calculate for the equation because that x value is the same x value for every point on the line. So if your x intercept is (5,0) then your line's equation would be x = 5.
Hope this helped :)
Happy holidays!!
• 5x-y is greater than or equal to 5 and y<5
• Fblpn,
5x-y >= 5 and y=5
If you change the first equation to slope y-intercept form
5x-y >= 5 add y to both sides
5x-y+y >= 5+y The y-y = 0 and disappears
5x >= 5+y And subtract 5 from both sides
5x-5 >= y Now reverse the sides and reverse the sign
y <= 5x-5 So we now the slope is 5 and y-intercept is (0,-5)
So graph that line (solid because it is also = to
and shade everything below the line since it is also <

The y<5 can be rewritten as
y=0x+5 So the slope is 0 (a horizontal line) that crosses the y axis at 5.
So graph that line (dashed line because it is not = to)
and shade below the line since it is <

I hope that helps make it click for you.
• how do I know to shade above or below the line
• If the inequality has a symbol of greater than or equal to or greater than you shade above the line. If the inequality has a less than or equal to or less than symbol you shade below the line.

HOPE THIS HELPS:)
• if I have something like y>-3 and the question says to 'graph the inequality in the coordinate plane'. Then what does the -3 signify/refer to when I put this inequality into slop intercept form to graph it......?
• -3 is the y-intercept. There is no slope (coefficient of x) so you know this is a straight horizontal line at -3. Since y>-3, any value above y=-3 would be a solution to the problem.
• how can i now if the equation is >= or just > i dont see the deference especially when we have the graph and we wont the equation
• If the line in the graph is solid then the inequality is >=.
If the line is dashed, then the inequality is just >.
If you watch the entire video, you will see toward the end that Sal changes the solid line into a dashed line since the 2nd example is for >.
• At , why is -x/2 the same thing as -1/2x?
• There is an invisible 1 in front of the x, so -x/2=-1x/2, then you can separate into two fractions, -1/2 *x/1, and dividing by 1 does not change anything, so you end up with -1/2 x.
• how can you tell what side you shade and which one not to shade I need help. Someone explain?
• Let's say you have an equation for an inequality, for instance y < 4x+3.
Draw the line on the graph (you can use the video of the graph since its the same equation except I'm using a less than instead of less than or equal to). Now you can do this problem a few ways. I think the easiest way at least for me is to simply pick a coordinate from any side of the line. You have the freedom to pick any point you can and just check if this inequality holds.

So, using this freedom let's pick (-2,2) which is on the left side of the line. Plug this into the equation: We get 2 < 4*(-2) +3, which we get this inequality 2 < -5.
Now, let's think is this correct?
No, it's not. So, that side of the line probably won't be shaded.

Continue using our freedom of choice, let's pick a point on the right side of the line. I like the point (2,0). So, plugging this into our equation gets us
0 < 4*2 +3. Is 0 < 11? Yes it is!

We can check this millions and millions of times on the right side and it turns out this inequality will hold!