Graphing Inequalities in Two Variables Investigation

Graphing Inequalities Investigation                                                        Name:  _______________________

Learning Goals: 

NC.M1.A-REI.12:  I can locate the region to a linear inequality in two variables, that represents the solutions of that inequality. Understand how the inequality sign effects the solution to the inequality.



a.  Go to

b.  Click on “start graphing”

c.  Follow the directions and answer the questions below.


1.    If the “keyboard” is not showing, click on the icon on the bottom left corner.  In the box in the upper left corner, type in the following equation:  y = x + 2.  You should see the line for this equation.  In the box below your equation, type in the following inequality:  y ≤ x + 2.  How is the graph of the inequality different than the graph of the equation?  (If unsure, toggle back and forth between the two statements). _________________________________________________________________

Why is the inequality shaded below the y-intercept of the line and not above it? (Hint:  look at the inequality sign)  ___________________________________________________________________________________________________________________

Locate the following points on the graph of y < x + 2:   (2, -1), (1, 0), (-2, -5).  All of these points are solutions of the inequality.  Find the point (1, 5).  This is not a solution.  From your observations, what might someone think the shading represents? _______________________________________________________________________________________________________

2.  Go to the first equation and change the = sign to ≥, so that the inequality reads y ≥ x + 2. How is the graph of this inequality different than the second one, y ≤ x + 2?  _________________________________________________________________

Why might someone think the first inequality is shaded above the y-intercept of the line on y ≥ x + 2?  ___________________________________________________________________________________________________________________

Name a point (x, y), not on the line that is a solution to the graph _________________


3.  Delete the 2nd inequality (y ≤ x + 2), and in the 1st box change the ≥ to > .  How did the graph change? (if you are not sure, keep changing the inequality back and forth until you see the difference?) ______________________________________

Why might the line be dashed instead of solid? What might the dashed line mean? (Hint:  Look at your new inequality. How is it different?) ________________________________________________________________________________________________ 

4.  Now change the inequality sign to < to read y < x + 2?   Now what does the graph look like?  __________________________

Name a point (x, y) within the shaded area that is a solution to the graph?  ______________________


5.  Summarize your findings:

A linear inequality with the symbol  ≥   has a ______________ line and is shaded ______________.

A linear inequality with the symbol  >  has a ______________ line and is shaded _______________.

A linear inequality with the symbol  ≤  has a _______________line and is shaded _______________.

A linear inequality with the symbol  <  has a ______________ line and is shaded _______________.



6.  Play around with the Desmos calculator to see if you can figure out what a system of linear inequalities looks like.  For example, you may want to type in the following:   y > x + 2   and y ≤ -2x + 1


a.  Where would the solution of the graphs be?


b.  How is this the same as what you just learned?


c.  How is it different?





1.    The equation contains a solid line, while the inequality has a shaded region. The line is shaded below the y-intercept because it has a less than or equal to inequality sign. The shaded region represents the solutions to the inequality.


2.    The inequality is shaded above the y-intercept. It is shaded above because the inequality sign is greater than or equal to. Solutions to the graph may vary.


3.    The line changes to dashed/dotted. The inequality sign is now just greater than and NOT equal to, so the dashed line must mean the points that make up that line are NOT part of the solutions set.


4.    Now the graph is dashed but shaded below the y-intercept of the line. Answers for the solution point may vary.


5.    Solid, above the y-intercept

Dashed, above the y-intercept

Solid, below the y-intercept

Dashed, below the y-intercept


6.    The solution set is located in the overlapping shaded region. It’s the same as what was learned because you must still look for the shaded region. It is different because it’s only the overlapping shaded region. (Answers may vary for this).

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