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Algebra 1

Course: Algebra 1 > Unit 7

Lesson 2: Graphing two-variable inequalities
Math
Algebra 1
Inequalities (systems & graphs)
Graphing two-variable inequalities
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Graphing inequalities (x-y plane) review

Google Classroom
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:
A coordinate plane with a graphed system of inequalities. The x- and y-axes both scale by two. There is a solid line representing an inequality that goes through the points zero, two and three, zero. The shaded region for the inequality is below the line.
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≤−24.
So, we put it in slope-intercept form:
4x+8y≤−248y≤−4x−24y≤−48x−3y≤−12x−3
Notice:
  • We shade below (not above) because y is less than (or equal to) the other side of the inequality.
  • 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.
A coordinate plane with a graphed system of inequalities. The x- and y-axes both scale by two. There is a solid line representing an inequality that goes through the points negative six, zero and zero, negative three. The shaded region for the inequality is below the line.
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:
−12x−4y<5−4y<12x+5y>−3x−54
Notice:
  • We shade above (not below) because y is greater than the other side of the inequality.
  • 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.
A coordinate plane with a graphed system of inequalities. The x- and y-axes both scale by two. There is a dashed line representing an inequality that goes through the points negative one, one point seven-five and zero, negative one point two-five. The shaded region for the inequality is above the line.

Example 3

We're given a graph and asked to write the inequality.
A coordinate plane with a graphed system of inequalities. The x- and y-axes both scale by two. There is a dashed line representing an inequality that goes through the plotted points zero, negative two and one, two. The shaded region for the inequality is above the line.
Looking at the line, we notice:
  • y-intercept is −2
  • Slope is ΔyΔx=41=4
The slope-intercept form of the inequality is
y ? 4x−2
where the "?" represents the unknown inequality symbol.
Notice:
  • The graph is shaded above (not below), so y is greater than the other side of the inequality.
  • 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:
y>4x−2
Want to see another example but in video form? Check out this video.

Practice

Problem 1
Which graph represents 8x−5y<3?
Choose 1 answer:

Want more practice? Check out this graphing exercise and this writing inequalities exercise

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  • hopper cool style avatar for user Owen
    Posted 5 years ago. Direct link to Owen's post “Can anybody give me some ...”
    Can anybody give me some tips on this subject, it's still kind of confusing. Thanks
    (15 votes)
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    • stelly green style avatar for user Faith Schultz
      Posted 3 years ago. Direct link to Faith Schultz's post “Hi! I know this is late a...”
      Incredible Answer
      Great Answer
      Good Answer
      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)
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    Posted a year ago. Direct link to Isaiah Boissssssssssssssseau's post “Am i the only human here?...”
    Am i the only human here? or is it all just bots?
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  • duskpin seedling style avatar for user yetzareli
    Posted 6 years ago. Direct link to yetzareli's post “How do you graph x>= -2 ,...”
    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)
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    • duskpin ultimate style avatar for user Pei-Hsin Lin 林培心 🌸
      Posted 6 years ago. Direct link to Pei-Hsin Lin 林培心 🌸 's post “x = -2 is a vertical line...”
      Good Answer
      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)
  • purple pi teal style avatar for user Mruthika
    Posted 4 years ago. Direct link to Mruthika's post “How does this help us in ...”
    How does this help us in the real world?
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    can you spin the block twice like there ain't nowhere to park?
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  • duskpin seed style avatar for user Lily500
    Posted 7 years ago. Direct link to Lily500's post “How do you graph with two...”
    How do you graph with two inequalities??
    (2 votes)
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  • blobby green style avatar for user sai melam
    Posted 3 years ago. Direct link to sai melam's post “Can anyone answer this fo...”
    Can anyone answer this for me:
    Choose the graph of inequality x > -2
    (4 votes)
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    • female robot grace style avatar for user loumast17
      Posted 3 years ago. Direct link to loumast17's post “to find the graph of an i...”
      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?
      (5 votes)
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    Why is Khan Academy so hard to learn? I don't feel that it is helping me in any way.
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  • female robot amelia style avatar for user lajachanaeminter
    Posted 7 years ago. Direct link to lajachanaeminter's post “so how do you determine t...”
    so how do you determine the y intercept when there isnt one ? i am really having a hard time with graphing can anyone help me ?
    (0 votes)
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    • mr pink green style avatar for user David Severin
      Posted 7 years ago. Direct link to David Severin's post “There is always a y - int...”
      There is always a y - intercept for linear functions, the only linear equations without a y intercept is a vertical line (x = #). So if you do not see the y-intercept you have to find it either by continuing the pattern until you find where it is or calculating it by using two points or one point and the slope. Two points requires calculating the slope (y2-y1)/(x2-x1) and putting into slope intercept or point slope form to find the y intercept.
      (8 votes)
  • winston default style avatar for user kharner01
    Posted 6 months ago. Direct link to kharner01's post “how do you know where to ...”
    how do you know where to test points and how do you put it into an equation
    (1 vote)
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