Now we are all set to apply the suggested steps in graphing direct inequality native the previous lesson. Let’s go over four (4) examples covering the different species of inequality symbols.

You are watching: Which inequality has a solid boundary line when graphed?

Example 1: Graph the direct inequality y>2x-1.

The first thing is come make certain that change y is by chin on the left side of the inequality symbol, which is the situation in this problem. Next is to graph the boundary line by momentarily transforming the inequality symbol to equality symbol.


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Graph the heat y>2x-1 in the xy axis utilizing your wanted method. Since the inequality price is just higher than “>” , and also not greater than or equal to “≥“, the boundary line is dotted or dashed. So here’s just how it must look therefore far.


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The last step is to the shade either over or below the boundary line. Indigenous the said steps, us were called to shade the top side of the boundary line if we have the inequality symbols >(greater than) or ≥(greater than or equal to). Constantly remember that “greater than” means “top”.


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To examine if your last graph of the inequality is correct, we have the right to pick any points in the shaded region. Because that this, let’s have actually the point (−1, 1).


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Evaluate the x and also y worths of the allude into the inequality, and see if the declare is true. In the point (−1,1), the values are x=-1 and y=1.


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2x-1 indicates 1>2(-1)-1, 1>-3 which is true" class="wp-image-119894" />

Since the test allude from the shaded region yields a true statement after ~ checking v the initial inequality, this reflects that our last graph is correct!

Example 2: Graph the linear inequality y \ge -x+2.

The change y is uncovered on the left side. That’s good! Notice, we have actually a “greater 보다 or same to” symbol. The “equal” element of the price tells us that the border line will certainly be solid. For this reason let’s graph the heat y=-x+2 in the Cartesian plane.


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Just prefer in instance 1, we will certainly shade the top portion the the border line due to the fact that we have a “greater than” case.


Verify if ours graph is correct by picking the point (4,2) in the shaded section, and evaluate the worths of x and also y that the suggest in the given linear inequality.


We have actually a true explain which provides us confident the our final graph that the inequality is correct as well.

Example 3: Graph the systems to the straight inequality \large{y .

Looking at the problem, the inequality price is “less than”, and also not “less than or same to”. Due to the fact that of this, the graph the the boundary line will certainly be damaged or dashed. In addition, “less than” method we will certainly shade the region below the line. That’s all there’s come it!

Here’s the graph of the border line \largey = 1 \over 2x - 1 .


I will leave it come you come verify that this is the correct graph by picking any type of test points native the shaded area and also check them versus the original linear equality.

Example 4: Graph the equipment to the direct inequality y \le - 2 \over 3x + 2.

Since we have actually gone end a few examples already, I believe that you can virtually work this the end in her head. You can impress your teacher by offering a brief solution as with this.

I see that the inequality symbol is “less than or equal to” ( ≤) which provides the boundary line solid. More so, the systems is listed below the boundary line due to the fact that of the “less than” element of it. Here’s the exactly graph of the inequality.


In the examples above, you have seen linear inequalities wherein the y-variables are always found ~ above the left side. You may also think of castle as direct inequalities in slope-intercept kind of a line.

X and also Y are on the exact same side of the inequality symbol

This time, we space interested in instances where the x and y variables are located on the exact same side of the inequality symbol.

We may contact them as direct inequalities in Standard Form. The complying with are 4 general instances where A, B, and C are just numbers or constants.


What we must do is come rewrite or manipulate the offered inequality such the the change y is required to remain on the left side. In various other words, we are going to resolve for y in terms of x. ~ doing so, we have the right to now apply the said steps in graphing straight inequality together usual.

Let’s walk over part examples.

Example 5: Graph the linear inequality in standard kind 4x + 2y .

Start fixing for y in the inequality by keeping the y-variable on the left, if the rest of the stuff are moved to the ideal side. Perform that by subtracting both sides by 4x, and also dividing v the whole inequality by the coefficient that y which is 4. Since we division by a optimistic number, the direction of the inequality symbol continues to be the same.


Since we have actually a “less than” symbol () and also not “less than or equal to” symbol (≤), the boundary line is going to be dotted or dashed.

Just in instance you forgot wherein to obtain the boundary line, readjust the inequality come equality price for the time being, that is, from y come y=-2x+4. Then graph the equation the the heat using any kind of of this methods.


So the next noticeable step is to decision which area to shade. Would it be above or listed below the border line? We will shade the bottom region of the border line because we have a “less than” situation after we reinvented the initial inequality problem into the type in which is the y is ~ above the left side.


We have the right to verify if we have graphed it appropriately by choosing any test points found in the shaded region. The best test point is the beginning which is the allude (0,0) because it is basic to calculate.


The test allude (0,0) method x=0 and also y=0. Advice these values in the transformed inequality or the original inequality to watch if you acquire a true statement.


It does work! therefore we have shaded the correct region which is listed below the dashed line.

Example 6: Graph the direct inequality in standard kind 3x - 6y \le 12.


To keep the variable y top top the left side, I would subtract both political parties by 3x and also then division the whole inequality by the coefficient that y i beg your pardon is 6.

REMEMBER: When dividing the inequality by a an adverse number, we must readjust or switch the direction of the inequality symbol.

See more: What Is The Gcf Of 36 And 54 ? Methods To Find Gcf Of 36 And 54

The “new” inequality will have actually a solid border line because of the symbol “≥” wherein it has actually the “equal ” component to it. In addition, due to the fact that y is “greater than” that method I will shade the an ar above the line.


You might additionally be interested in:

Solving direct Inequalities

Steps on exactly how to Graph direct Inequalities

Graphing solution of linear Inequalities

Solving compound Inequalities


MATH SUBJECTSIntroductory AlgebraIntermediate AlgebraAdvanced AlgebraAlgebra word ProblemsGeometryIntro come Number TheoryBasic mathematics Proofs
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