Up

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.

Systems-of-Equations-&-Inequalities-1

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:

Systems-of-Equations-&-Inequalities-2

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:

Systems-of-Equations-&-Inequalities-3

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:

Systems-of-Equations-&-Inequalities-4

Below you can download some free math worksheets and practice.


Downloads:
4421 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:

Systems-of-Equations-and-Inequalities-Graphing-systems-of-inequalities-easy

Watch below how to solve this example:

 

Downloads:
2471 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:

Systems-of-Equations-and-Inequalities-Graphing-systems-of-inequalities-medium

Watch below how to solve this example:

 

Downloads:
2051 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:

Systems-of-Equations-and-Inequalities-Graphing-systems-of-inequalities-hard

Watch below how to solve this example:

 
 
 

Facebook PageGoogle PlusTwitterYouTube Channel

Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Equations
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Exponents
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
Factoring
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Slope
Graphing absolute value equations
Percents
Percent of change
Markup, discount, and tax
Polynomials
Adding and subtracting
Dividing
Multiplying
Naming
Quadratic Functions
Completing the square by finding the constant
Graphing
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Inequalities
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Dividing
Equations
Multiplying
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Equations
Multiplying and dividing
Simplifying and excluded values
Systems of Equations and Inequalities
Graphing systems of inequalities
Solving by elimination
Solving by graphing
Solving by substitution
Word problems