Graphing systems of inequalities

There are a lot of little elements that you need to know in order to graph a system of inequalities. Much of it is the same as graphing a line, but we will go through all of it one step at a time.

Let’s start out with this system:

$$x + y \geqslant - 1$$
$$y > - 4x + 2$$

Step One: Make sure both inequalities are solved for “y.” This means that “y” must be by itself. The second inequality is ok, but we have to change the first one.

$$x + y \geqslant - 1$$
x          –x
$$y \geqslant - 1 - x$$
or $$y \geqslant - x - 1$$

Our system now looks like this:

$$y \geqslant - x - 1$$
$$y > - 4x + 2$$

Step Two: Take one inequality at a time and graph. Let’s take $$y \geqslant - x - 1$$ and split this step into two:

Remember:
$$y = mx + b$$
m= slope
b= y-intercept

Your “starting point” is the y-intercept. Find this value on the y-axis and plot a point.  So, our starting point is at -1 on the y-axis.

To find more points, we have to use the slope, which is $$\Large \frac{{rise}}{{run}}$$. The slope in this example is $$\Large \frac{{ - 1}}{1}$$ which means down one, right one. So, let’s go back to our y-intercept and plot some more points.

Step Three: Connect the points with a SOLID LINE if the inequality is $$\leqslant$$ (less than or equal to) or $$\geqslant$$  (greater than or equal to) and a DOTTED LINE if the inequality is (greater than). This first example is a solid line. So we have:

Now, we have to do this all over again with the second inequality!

$$y > - 4x + 2$$

This time our y-intercept is +2 and our slope is $$\frac{{ - 4}}{1}$$ which means down 4 and right 1. It is also a DOTTED LINE. So we now have:

Step Four: We have to shade in part of our graph since there is more than one value that will work in our system of inequalities. For (greater than) or $$\geqslant$$ (greater than or equal to), we shade above the line (think of the line as a slide and that’s “above”). In our example, both inequalities are the “above” inequalities so our shading must be above BOTH lines. Our final graph should look like:

4932 x

Sketch the solution to each system of inequalities.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Watch below how to solve this example:

2875 x

Sketch the solution to each system of inequalities.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Watch below how to solve this example:

2333 x

Sketch the solution to each system of inequalities.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Watch below how to solve this example:

Geometry

Circles
Congruent Triangles
Constructions
Parallel Lines and the Coordinate Plane
Properties of Triangles

Algebra and Pre-Algebra

Beginning Algebra
Beginning Trigonometry
Equations
Exponents
Factoring
Linear Equations and Inequalities
Percents
Polynomials