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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:
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 left parenthesis, 0, comma, 0, right parenthesis is a solution because it is in the shaded part, but the point left parenthesis, 4, comma, 4, right parenthesis 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 4, x, plus, 8, y, is less than or equal to, minus, 24.
So, we put it in slope-intercept form:
4x+8y248y4x24y48x3y12x3\begin{aligned}4x+8y&\leq -24\\\\ 8y&\leq -4x-24\\\\ y&\leq-\dfrac{4}{8}x-3\\\\ y&\leq-\dfrac{1}{2}x-3 \end{aligned}
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 minus, 12, x, minus, 4, y, is less than, 5.
So, we put it in slope-intercept form:
12x4y<54y<12x+5y>3x54\begin{aligned}-12x-4y&< 5\\\\ -4y&< 12x+5\\\\ y&>-3x-\dfrac{5}{4} \end{aligned}
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 start color #7854ab, minus, 2, end color #7854ab
  • Slope is start fraction, delta, y, divided by, delta, x, end fraction, equals, start fraction, 4, divided by, 1, end fraction, equals, start color #e07d10, 4, end color #e07d10
The slope-intercept form of the inequality is
y, space, question mark, space, start color #e07d10, 4, end color #e07d10, x, start color #7854ab, minus, 2, end color #7854ab
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, is greater than, 4, x, minus, 2
Want to see another example but in video form? Check out this video.

Practice

Problem 1
Which graph represents 8, x, minus, 5, y, is less than, 3?
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • hopper cool style avatar for user Owen
    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
      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).
      (67 votes)
  • duskpin seedling style avatar for user yetzareli
    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.
    (3 votes)
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    • duskpin ultimate style avatar for user Pei-Hsin Lin 林培心 🌸
      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!
      (17 votes)
  • duskpin seed style avatar for user Lily500
    How do you graph with two inequalities??
    (3 votes)
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  • purple pi teal style avatar for user Mruthika
    How does this help us in the real world?
    (5 votes)
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  • starky tree style avatar for user Abner Lopez
    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?
    (2 votes)
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  • blobby green style avatar for user Isaiah Boissssssssssssssseau
    Am i the only human here? or is it all just bots?
    (2 votes)
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  • female robot amelia style avatar for user lajachanaeminter
    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
      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.
      (7 votes)
  • blobby green style avatar for user Schweitzer, Isaac
    So, from what I'm getting, you can only express the solution to systems of inequalities by shading the parts that they cover, right? You cant use coordinates? It's not like a normal system where you have an actual coordinate to show the solution.
    (3 votes)
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    • starky tree style avatar for user Jennifer
      Hi! This answer is probably late, but 😅basically, yeah, you can't express a line using inequalities because a line goes on FOREVER, meaning there are an INFINITE amount of coordinates. So, yes, you'd have to shade in the area the line covers.
      Hope that helps!
      (0 votes)
  • blobby green style avatar for user Jennifer Richardson
    I'm confused and how to know when to change the inequality sign. I thought we would change it when we divide but I noticed that isn't always the case. Can you help?
    (2 votes)
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  • blobby green style avatar for user sai melam
    Can anyone answer this for me:
    Choose the graph of inequality x > -2
    (1 vote)
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    • female robot grace style avatar for user loumast17
      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?
      (3 votes)