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Graphing linear inequalities combines the skills you already know from graphing linear equations with the skills you’ve gained from understanding the basics of linear inequalities.

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Just like the lesson on linear inequalities, we have to consider whether we have strict (< or >) or weak ( or ) inequality.

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Strict inequality means that we will end up drawing a dashed line to indicate that the line itself is not part of the region.

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Weak inequality means that we will draw a solid line, indicating that the line is part of the region.

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Also, notice how I’ve been saying regions, that’s because the graph of a linear inequality is actually a whole region of our coordinate plane!

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For instance, take the linear inequality x>4. This means that the only restriction we have is that our x value has to be larger than 4 and y can be whatever it wants. This would be graphed with a dashed line (since it is strict inequality).

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Also, notice how on this graph, we see that the line x=4 is a vertical line, which is dashed! This is identical to how you have graphed lines in the past.

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For weak inequality, we can graph a solid line and then determine the region to shade in.

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For instance, take the inequality y2x-3. If we were simply graphing this as an equation, we would be graphing the line y=2x-3 and slope-intercept form tells us that the slope of this line is 2 and it has a y-intercept at -3.

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Now that we have the graph of the line, we must determine what our inequality actually says. Since y is less than or equal to this line, we must shade in the region that is underneath the line! This is graphed in the following way:

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Let’s consider another strict inequality. Take the linear inequality y>-x+2and use the same procedure we outlined above.

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First, graph the line y=-x+2, which has a slope of -1 and a y-intercept of 2. This looks like the following graph.

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Now, consider the region to shade in. Since y is strictly greater than this line, we must change our line into a dashed line and shade the region above it!

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Just like this, we have our linear inequality totally graphed. It’s practically identical to graphing linear equations!

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However, the meaning behind the regions of these linear inequalities is a little unclear. It’s fairly intuitive to see that if y is greater than something it needs to be above and if it is less than something, then it’ll be below. What does this actually mean?

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You can imagine these shaded regions as all of the values of x and y that make the inequality true. This means that the region is actually just representing a set of points where our inequality is true!

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For example, consider the last inequality that we graphed. Once we had graphed y>-x+2, we could choose any point in our shaded region and it should satisfy the inequality. Take the point (2, 1) and substitute these values into the inequality.

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This shows us that 1>-(2)+2, which is basically saying that 1>0. Obviously this is true!

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However, notice what happens when we plug in the point (2, 0) that is on the dashed line. This shows us that 0>-(2)+2, which is saying that 0>0, which is not true!

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This is why we drew a dashed line to begin with. If our inequality had been y-x+2, then that point on our line would have actually worked.

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At this point, you should feel comfortable and confident in graphing linear equations and linear inequalities. Be sure to remember the difference between strict and weak inequality for graphing purposes in particular and understand that when you are shading a region, you’re actually defining a set of points that make your inequality true!

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