Previously, you learned exactly how to graph a solitary linear inequality top top the xy plane. In this lesson, we will deal with a system of linear inequalities. The word “system” suggests that we are going come graph two or more linear inequalities simultaneously. The systems to the system will it is in the area or an ar where the graphs of all direct inequalities in the device overlap.

You are watching: Which inequality will have a shaded area below the boundary line?

NOTE: In order come be successful in graphing linear inequalities, you room expected come know exactly how to graph a heat on the xy-plane. Otherwise, take a minute to review the material.

Steps how to Graph system of straight Inequalities

Step 1:Graph every linear inequality in the mechanism on the very same xy axis. Mental the vital steps once graphing a straight inequality:

Isolate the “y” change to the left the the inequality.If the symbols room > and also ≥, us shade the area over the boundary line utilizing dashed and solid lines, respectively.On the various other hand, if the icons are and ≤, we shade the area listed below the border line making use of dashed and solid lines, respectively.

Step 2:Shade the region where every the locations of the straight inequalities intersect or overlap. If over there is no region of intersection, we say that the system has no solution.

Let’s walk over some examples to illustrate the procedure.

Examples of Graphing system of direct Inequalities


The good thing about the given problem is that all y-variables are already on the left next of the inequality symbol. In this form, us can conveniently determine what area to shade with recommendation to the boundary line.

Graph the an initial inequality yx −1. Due to the fact that we have actually a “less than or same to” symbol, the border line will be solid and we shade the area listed below the line.

Graph the 2nd inequality y . The price is simply “less than” therefore the boundary line will certainly be dashed or dotted, and we shade the area listed below it.

The final solution to the system of direct inequalities will certainly be the area where the two inequalities overlap, as displayed on the right.

We speak to this solution area as “unbounded” since the area is actually extending forever in downward direction.

You will see an instance of “bounded” area on the next example.



The system includes three inequalities, that method we are going come graph three of them. Notice, every the inequality symbols have actually an “equal to” component. This tells us that every the border lines will certainly be solid.

Here is the graph the the first inequality
where the border line is solid and the shaded area is found listed below it.
As you have the right to see, the shaded areas of the three straight inequalities overlap ideal in the middle section.

We speak to this mechanism “bounded” due to the fact that the an ar where all services lie space enclosed through the 3 sides comes from the boundary lines of the direct inequalities.

When i look at the 3 inequalities contained in the system, there are three points that I should consider:

Rewrite the first inequality x + 2y such the the “y” change is alone ~ above the left side. If you work-related this out properly to isolate “y“, this inequality is tantamount to the expression .The inequality y>–1will have actually a horizontal boundary line.The inequality x≥–3will have a vertical boundary line.

If girlfriend need aid how come graph vertical and horizontal lines, inspect it the end on this different lesson.

Now to be are prepared to graph each among them.

The graph of will certainly be a dotted boundary line through shaded the area uncovered under it.
The graph of y > –1is a dashed horizontal heat passing v the y-intercept at -1 through the shaded area above it.
The graph that x ≥ –3is a solid line passing with the x-intercept at -3 with the shaded area come its right.
The systems to this mechanism is the common area where all 3 inequalities intersect.

This is likewise a “bounded system” whereby the solution an ar are fastened by two dashed line segments and also one solid heat segment.

Both that the inequalities need some rewriting so the the change “y” is located on the left side by itself.

Here’s for the first one. Make certain to move the direction the the inequality prize whenever you divide the inequality by a negative number.

And here’s the rewriting of the second inequality. This time we don’t divide the inequality by a an adverse number that’s why the orientation or direction of the inequality symbol remains the same.

Let’s go ahead and graph them.

The graph that y > –2x + 1 is a dotted or “broken” line having actually the shaded area above it.
The graph of y ≤ –2x − 3is a hard line with the shaded area under it.

Since the 2 shaded locations don’t crossing or overlap, this tells united state that the provided system the inequalities has NO SOLUTION.

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You may have also observed the the boundary lines have actually equal slopes , both m = –2, which implies that they room parallel lines, and therefore won’t intersect.

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