Graphing Inequalities: How to Graph y - x 3

When working with inequalities in mathematics, it's essential to understand how to graph them accurately. This guide will walk you through the process of graphing the inequality y - x 3. As we'll see, the steps are similar to those for graphing equations, but with some key differences, particularly in shading.

Understanding the Inequality

The given expression y - x 3 is indeed an inequality. Our goal is to determine all points [x, y] that satisfy the inequality. In this case, we're dealing with a linear inequality, which can be plotted as a line with a designated shading area representing the solution set.

Step 1: Converting to Slope-Intercept Form

To make graphing easier, we'll convert the inequality to slope-intercept form:

y - x 3

Adding x to both sides:

y x 3

This is now in the form y mx b, where m is the slope and b is the y-intercept. Here, m -1 and b 3.

Step 2: Graphing the Line

Next, we plot the line y -x 3:

The y-intercept is b 3, so we start at point (0, 3).

To find another point, we use the slope. Since the slope is -1, we go down 1 unit and to the right 1 unit from (0, 3). This gives us the point (1, 2).

Plot point (0, 3) and (1, 2), and draw a dashed line through these points. The line is dashed because the inequality is strict (no equality).

Step 3: Testing a Point

To determine which side of the line to shade, we test a point that is not on the line. A convenient choice is the origin (0, 0).

Substitute x 0 and y 0 into the inequality:

y - x 3 becomes

0 - 0 3 which is not true.

Therefore, the origin (0, 0) does not satisfy the inequality. We should shade the side of the line that does not include (0, 0).

Step 4: Shading the Solution Set

Since the point (0, 0) is not part of the solution set, we shade the half-plane below the line y -x 3. This shaded area represents all points that satisfy the inequality y - x 3.

Conclusion

Graphing linear inequalities like y - x 3 involves converting to slope-intercept form, plotting the line, and then testing a point to determine the correct side to shade. This method applies to many inequalities and helps visualize the solution set more clearly.

Feel free to practice with different inequalities to deepen your understanding and proficiency in graphing.