Main content

## Integrated math 1

### Course: Integrated math 1 > Unit 7

Lesson 2: Graphing two-variable inequalities- Intro to graphing two-variable inequalities
- Graphing two-variable inequalities
- Graphs of inequalities
- Two-variable inequalities from their graphs
- Two-variable inequalities from their graphs
- Intro to graphing systems of inequalities
- Graphing systems of inequalities
- Systems of inequalities graphs
- Graphing inequalities (x-y plane) review

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Graphing inequalities (x-y plane) review

We graph inequalities like we graph equations but with an extra step of shading one side of the line. This article goes over examples and gives you a chance to practice.

The graph of a two-variable linear inequality looks like this:

It's a line with one side shaded to indicate which $x$ -$y$ pairs are solutions to the inequality.

In this case, we can see that the origin $(0,0)$ is a solution because it is in the shaded part, but the point $(4,4)$ is not a solution because it is outside of the shaded part.

*Want a video introduction to graphing inequalities? Check out this video.*

### Example 1

We want to graph $4x+8y\le -24$ .

So, we put it in slope-intercept form:

Notice:

- We
**shade below**(not above) because is less than (or equal to) the other side of the inequality.$y$ - We
**draw a solid line**(not dashed) because we're dealing with an "or equal to" inequality. The solid line indicates that points on the line are solutions to the inequality.

*Want to see another example but in video form? Check out this video.*

### Example 2

We want to graph $-12x-4y<5$ .

So, we put it in slope-intercept form:

Notice:

- We
**shade above**(not below) because is greater than the other side of the inequality.$y$ - We
**draw a dashed line**(not solid) because we aren't dealing with an "or equal to" inequality. The dashed line indicates that points on the line are not solutions of the inequality.

### Example 3

We're given a graph and asked to write the inequality.

Looking at the line, we notice:

-intercept is$y$ ${-2}$ - Slope is
$\frac{\mathrm{\Delta}y}{\mathrm{\Delta}x}}={\displaystyle \frac{4}{1}}={4$

The

*slope-intercept form*of the inequality iswhere the "?" represents the unknown inequality symbol.

Notice:

- The graph is
**shaded above**(not below), so is greater than the other side of the inequality.$y$ - The graph has a
**dashed line**(not solid), so we aren't dealing with an "or equal to" inequality.

Therefore, we should use the greater than symbol.

The answer:

*Want to see another example but in video form? Check out this video.*

## Want to join the conversation?

- Can anybody give me some tips on this subject, it's still kind of confusing. Thanks(16 votes)
- Hi! I know this is late and that you 100% won't see this comment, BUT I like to help and LOVE math. So, here's my tip: when looking to find the graph of an inequality, look at inequality sign first. If it has a line directly below it, it is deemed inclusive, indicating a solid line. If there is no line under the inequality sign, it is deemed non-inclusive, indicating a dashed line. Then, look at the the y term--not y-intercept. Take note of it's value. If it is a negative you are going to want to flip the direction of the sign. For instance, if you have the linear inequality -5y>8x+1, you might initially assume that the solutions to the inequality will be represented by shading the half plane that is above the y-intercept 1, but this is incorrect. In order to isolate the y variable we have to divide it by -5, along with other expression of the inequality (8x+1). Hence, we FLIP the original greater than sign (>) to a less than sign (<), which changes the entire format of the graph (or at least the solutions to the problem).(88 votes)

- Am i the only human here? or is it all just bots?(8 votes)
- There are lots of humans here. We're not bots.(20 votes)

- How do you graph x>= -2 , and why do you graph it vertically? and how do you know which side to shade? thank you, this is so confusing.(1 vote)
- x = -2 is a vertical line. It contains all points on the xy-plane where the value of x is -2.

To graph x ≥ -2, you have to know that ≥ is the**greater than or equal to**symbol.

The**equal**part means you'll need to use a**solid line**on the boundary itself (x = -2).

The**greater than**part means you'll need to shade the side of the line that has values of x that are**more than**-2. On an x-axis that is scaled and numbered properly, all the numbers more than -2 are clearly labeled**on the right**side of the vertical line.

That's how you know which side to shade!(20 votes)

- How does this help us in the real world?(10 votes)
- if you wanna be a financialist(3 votes)

- can you spin the block twice like there ain't nowhere to park?(10 votes)
- How do you graph with two inequalities??(3 votes)
- Just plot both lines on the graph and make sure to use the right y-intercept and if it's not an equal to sign make the line dotted(10 votes)

- Can anyone answer this for me:

Choose the graph of inequality x > -2(5 votes)- to find the graph of an inequality it is just like finding the graph of en equation. So first pretend it is x = -2

Now, the two extra steps are look at if it is just greater than or less than, or if it is also equal to. if it is greater than or less than the line of the graph is dashed. if it is greater than or equal to OR less than or equal to than it is a solid line like in a normal equation.

The second step is then to find where you shade in. With a linear equation it's super easy. If this were y > -2 you would shade above the line, so on the positive side. if it were y < -2 you would shade below the line, so on the negative side. So look at yours, check if you want to shade on the positive side or neagative side of the line, then determine which is which.

Can you handle it from there?(6 votes)

- Why is Khan Academy so hard to learn? I don't feel that it is helping me in any way.(3 votes)
- Yo I don't know if it's just me but Khan Academy is the only reason I passed all my math classes. It can be hard but with practice, it will get easier don't give up on Khan Academy!(10 votes)

- -6a-2<-74 on a line chart. looking for how to solve(2 votes)
- First, you are in the wrong lesson. The lessons you need are at:

1) Solving 2-step inequalities: https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-variables-expressions/cc-7th-two-step-inequalities/v/solving-inequalities

2) Plotting / graphing an inequality: https://www.khanacademy.org/math/pre-algebra/pre-algebra-equations-expressions/pre-algebra-greater-than-less-than/v/plotting-inequalities-on-a-number-line

Hope this helps.(3 votes)

- okay I know this is a dumb question but I need to confirm if an inequalitie is divide/multiply by a negative number the symbol ( <,> ) is flipped even if the negative number is the one being divided/multiple correct?(1 vote)
- Yes it sounds like you have the right concept, If the coefficient of the variable is negative, then you have to either divide or multiply by a negative to isolate the variable, and this causes the inequality to flip also.(5 votes)