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Algebra (all content)

Course: Algebra (all content) > Unit 6

Lesson 3: Graphing two-variable inequalities
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Algebra (all content)
Two-variable inequalities
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 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+8y≤−248y≤−4x−24y≤−48x−3y≤−12x−3\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:
−12x−4y<5−4y<12x+5y>−3x−54\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
  • Current
Which graph represents 8, x, minus, 5, y, is less than, 3?
Choose 1 answer:

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

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