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Your Step-by-Step Guide to Graphing Linear Inequalities

graphing linear inequalities: Calculator, eyeglasses, magnifying glass, pen and papers on a wooden surface

Graphing linear inequalities is similar to graphing linear equations but with the added steps of picking a solid or dotted line and determining what part of the coordinate plane to shade in order to make the inequality a true statement.

To review, a linear inequality is any statement that includes a symbol other than the equal sign. Greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤), are all examples of these inequality signs. A statement that contains one of these inequality symbols makes it a linear inequality.

Here are the steps to graphing a linear inequality.

1. Rearrange the Linear Inequality Similar to a Linear Equation

First, rearrange the linear inequality so that the y-variable is on one side. This is very similar to if you were graphing the line by itself.

You want the equation of the line to be in a y = mx + b format. Don't worry about the actual inequality symbol yet — the first step is graphing the straight line. Let's use the inequality y - 2 > 4x as an example. You would start by rearranging it such that it reads, y > 4x + 2. Then you graph it just as you would a linear equation.

2. Decide If It's a Strict Inequality

The second step is determining if the inequality is strict. This will show whether to use a solid line or a dashed line for the boundary line. The boundary line divides the plane to show which areas contain all the points that satisfy the inequality.

If the linear inequality uses ">" or "<," it is strict and will use a dashed line. This means that any point on the dashed line does not satisfy the inequality.

If you see that it uses "≥" or "≤," it is not strict and will use a solid line. The solid line means that any of the ordered pairs on the graph that fall exactly on the line will satisfy the inequality, and therefore is part of the solution set. Substituting those values for x and y in the original inequality will make a true statement. In other words, it will satisfy the linear inequality.

y > 4x + 2 uses a “>” symbol, so it is a strict inequality that requires a dotted line.

graphing linear inequalities: Graph showing a strict inequality

3. Determine the Shaded Region

Lastly, when graphing the linear inequality, you need to pick one side of the boundary line to shade. The shaded area contains all the points that satisfy the original inequality.

So how do you figure out which half-plane to shade? The easiest way is to pick a test point, usually (0, 0), and substitute it into the inequality. If it satisfies the linear inequality, shade the side that contains the test point. If it does not, leave the side that contains the test point blank and shade the other side.

graphing linear inequalities: Graph with a linear inequality with one side shaded

Here, substitute (0, 0) into the original linear equation.

Linear equation

Linear equation

Linear equation

This test-point does not satisfy the linear equality, so we choose to leave the side that contains (0, 0) blank and instead, shade the other side.

Tips for Graphing Linear Inequalities

If you see a statement with an inequality symbol (>, <, >=, <=), follow the three steps:

  1. Rearrange the linear equality
  2. Draw the dotted or solid line
  3. Choose which side to shade.

If you remember the tips for graphing a linear equation and add in these steps, you’ll be on your way in no time.

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