CCSS Math: HSA.REI.D.12
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 xx-yy pairs are solutions to the inequality.
In this case, we can see that the origin (0,0)(0,0) is a solution because it is in the shaded part, but the point (4,4)(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+8y244x+8y\leq -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 yy 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.
Want to see another example but in video form? Check out this video.

Example 2

We want to graph 12x4y<5-12x-4y< 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 yy 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.

Example 3

We're given a graph and asked to write the inequality.
Looking at the line, we notice:
  • yy-intercept is 2\purpleD{-2}
  • Slope is ΔyΔx=41=4\dfrac{\Delta y}{\Delta x}=\dfrac{4}{1}=\goldD{4}
The slope-intercept form of the inequality is
y ? 4x2y~?~\goldD{4}x\purpleD{-2}
where the "?" represents the unknown inequality symbol.
Notice:
  • The graph is shaded above (not below), so yy 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>4x2y>4x-2
Want to see another example but in video form? Check out this video.

Practice

Problem 1
Which graph represents 8x5y<38x-5y< 3?
Choose 1 answer:
Choose 1 answer:

Loading