Graphing Inequalities

 Mathematicians, like magicians, artists, and other masters of the visual arts, communicate best when they accompany the abstractions they communicate with visual aids. It is a long-established fact that the vast majority of people are visual learners. It is easy to graph simple linear equations on a number line. However, that isn’t the only way to visually depict a mathematical expression. By using a coordinate plane, it is again possible to plot the points of a linear expression. The coordinate plane is also useful for graphing inequalities. If the thought of graphing unequal equations sounds intimidating, breathe a sigh of relief. Although it is different from graphing expressions on a number line, it is every bit as simple.

Think back to how you plotted a linear expression on your number line. Say the problem was x < 6. What did you do to solve it? You drew a number line, marked the 6 with an open circle to indicate that the 6 wasn’t part of the answer, and then you shaded everything that was to the left to show that those numbers were less than 6. You’ll find that graphing a linear inequality with two variables is quite similar.

When graphing inequalities, we use the x and y coordinate plane instead of a number line. A coordinate plane is essentially another number line, a two-dimensional number line with two lines that intersect at a 90° angle at both lines’ point of zero, called the origin. The vertical line is known as the y-axis, and the horizontal line is the x-axis. These axes divide the coordinate plane into four quadrants that, beginning in the upper right quadrant and proceeding counter-clockwise, are known as quadrants 1, 2, 3, and 4.

Every intersection on the coordinate plane represents a particular ordered pair of (x,y) points. Points on the x-axis to the right of zero are positive numbers, and points to the left of zero represent negative numbers. On the y-axis, points above zero indicate positive numbers and points below zero indicate negative numbers.

Let’s start with an easy inequality. Graph y ≤ 2x + 4. Our goal is to find all the ordered pairs of x and y points that satisfy this expression. We’re really looking at two statements here, that y is = to 2x+4 and also, that y < 2x + 4. So, first you’d graph the straight line y = 2x + 4.

There are two ways to demonstrate the actual inequality contained in this expression. One, use a pencil to shade in the area of the solution. You’ve indicated the equals part with the line you’ve already graphed. Now, you just need to shade to one side or the other of the line to indicate the less than portion of the graph. If y is less than your line, you’ll want to shade below it.

Your second option is to use test points, and to plot them on your graph. Since you’ve already graphed what y is equal to, you now put your test numbers into the equation as x, and solve to find out what y is less than.  No matter what your test numbers, you’ll find that your y values will all be less than those plotted on your equals to line.

If the equation you were solving were not ≤, but simply <, you would indicate this by using a dotted line on the graph. The dotted line excludes itself from the equation much like the open circle on a number line