Linear Inequalities in Two Variables

5.5 - Lesson - Graphing Linear Inequalities Video Lesson
13:13 — by MrJSaintGermain

Tip: This isn't the place to ask a question because the teacher can't reply.

Key Questions

• The best way (when possible!) is to express the inequality bringing all the terms involing a variable on the left, and all the terms involving the other variable on the right. In the end, you'll have an inequality of the form $y \setminus \le f \left(x\right)$ (or $y \setminus \ge f \left(x\right)$), and this is easy to graph, because if you can draw the graph of $f \left(x\right)$, then you'll have that $y \setminus \le f \left(x\right)$ represents all the area under the function $f$, and $y \setminus \ge f \left(x\right)$, of course, the area over the function.

For example, consider the inequality
$y {x}^{2} < - y + 3 x$
In this form, it would be very hard to say which points satisfy the inequality, but with some manipulations we obtain
$y {x}^{2} + y < 3 x$
$y \left({x}^{2} + 1\right) < 3 x$
$y < \setminus \frac{3 x}{{x}^{2} + 1}$

Now, the graph of $\setminus \frac{3 x}{{x}^{2} + 1}$ is easy to draw, and the inequality is solved considering all the area below the graph, as showed:

graph{y<3x/{x^2+1} [-10, 10, -5, 5]}

Note that if you have $y < f \left(x\right)$ the graph of the function $f$ is not included, since it represents the points for which $y = f \left(x\right)$.

• You check by using a point in your range.
Take one point in the range (the shaded area) and check it again with your given equation.

EX

$y > 2 x + 4$ and $\left(1 , 7\right)$ was your chose coordinate

$7 > 2 + 4$
$7 > 6$

That is true, so your graph is correct.

If you still doubt yourself, you can always check more points.

• The solution set of a single linear inequality is always a half-plane, so there are infinitely many solutions.

I hope that this was helpful.

See below.

Explanation:

Shading for absolute inequalities means shading over the solution region. The easiest way to determine shading for linear inequalities in two variables is to choose a test point and plug into the original inequality to determine if one should shade above or below the line.

1. Imagine that the inequality sign is an equal sign and graph the equation on a coordinate plane.
2. If the inequality symbol is a $\le$ or $\ge$, then draw a solid line connecting the points. If the inequality symbol is a $<$ or $>$, then draw a dashed line connecting the points.
3. Choose a test point above the line and one that is below the line. Plug each ordered pair into the original inequality and one should be a false statement and one should be a true statement.
4. Shade the region with the true statement.

For example: Graph and shade $y \ge 3 x + 5$

Graph: $y = 3 x + 5$

Choose a test point above the line $\left(- 5 , 5\right)$ and plug into the original inequality.

$5 \ge 3 \left(- 5\right) + 5$

$5 \ge - 15 + 5$

$5 \ge - 10 \to$ True Statement

Choose a test point below the line $\left(0 , 0\right)$ and plug into the original inequality.

$0 \ge 3 \left(0\right) + 5$

$0 \ge 0 + 5$

$0 \ge 5 \to$ False Statement

So we want to shade above the line:

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