Step-by-step explanation:
The graph of a two-variable linear inequality looks like this:
It's a line with one side shaded to indicate which xxx-yyy pairs are solutions to the inequality.
In this case, we can see that the origin (0,0)(0,0)left parenthesis, 0, comma, 0, right parenthesis is a solution because it is in the shaded part, but the point (4,4)(4,4)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 4x+8y\leq -244x+8y≤−244, x, plus, 8, y, is less than or equal to, minus, 24.
So, we put it in slope-intercept form:
\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}
4x+8y
8y
y
y
≤−24
≤−4x−24
≤−
8
4
x−3
≤−
2
1
x−3
Notice:
We shade below (not above) because yyy 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 -12x-4y< 5−12x−4y<5minus, 12, x, minus, 4, y, is less than, 5.
So, we put it in slope-intercept form:
\begin{aligned}-12x-4y&< 5 -4y&< 12x+5 y&>-3x-\dfrac{5}{4} \end{aligned}
−12x−4y
−4y
y
<5
<12x+5
>−3x−
4
5
Notice:
We shade above (not below) because yyy 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:
yyy-intercept is \purpleD{-2}−2start color purpleD, minus, 2, end color purpleD
Slope is \dfrac{\Delta y}{\Delta x}=\dfrac{4}{1}=\goldD{4}
Δx
Δy
=
1
4
=4start fraction, delta, y, divided by, delta, x, end fraction, equals, start fraction, 4, divided by, 1, end fraction, equals, start color goldD, 4, end color goldD
The slope-intercept form of the inequality is
y~?~\goldD{4}x\purpleD{-2}y ? 4x−2y, space, question mark, space, start color goldD, 4, end color goldD, x, start color purpleD, minus, 2, end color purpleD
where the "?" represents the unknown inequality symbol.
Notice:
The graph is shaded above (not below), so yyy 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
y>4x-2y>4x−2y, is greater than, 4, x, minus, 2
