Students have learned that a line represents all points that satisfy the equation that defines it. All (x, y) points that satisfy y = 2x – 1 for example, are on the line defined by the equation. Whether a point like (0, 5) is on this line can be tested by substituting 0 for x and 5 for y in the equation; it is quickly seen that the point does not satisfy the equation since the result is 5 = –1, which is clearly untrue. What is true, however is that 5 > –1, so it can be said that the point (0,5) satisfies the inequality y > 2x – 1. In this lesson, students learn how to represent inequalities graphically.
This lesson establishes the basics which is built upon in the next unit, in which students learn to graph systems of inequalities.
Half Planes and Inequalities. Earlier in this unit students learned that the graphs for horizontal and vertical lines are in the form y = b and x = a for horizontal and vertical lines, respectively. (If b and a = 0, the graphs are the two axes.) These equations are often quickly forgotten, so this lesson comes in handy to remind them about the graphs, and to start the discussion on inequalities and half planes.
I draw the graph below on the board.
“Somebody please tell me the equation of this line.” This should be fairly fresh in most minds, so I’ll get a few hands and someone will tell me it is x = 3.
“Now this line divides the plane of the graph into two regions, that we call “half-planes”. The equation x = 3 is a boundary line of the two half-planes. Any point to the right of this vertical line is greater than 3. So the half-plane to the right of the line, including the boundary, satisfies the inequality x ≥ 3.” I shade the area to the right of the line.
“If I shaded the area to the left of the line and included the boundary, what would be the inequality?”
Students pick up on this fairly quickly and answer x ≤ 3.
“If I did not want to include the boundary line, we would indicate the boundary with a dotted line. So the graph of x > 3 would look like this.”
I do the same type of demonstration for x < 3, and then for horizontal lines, drawing a graph of y =5. “If I go above the line and include the boundary line, what is the inequality?” Answer: y ≥ 5. “How about without the boundary and shade below the line?” I ask, indicating that now there is a dotted line. Answer: y < 5.
Graphing Inequalities for Any Line. I move now to lines that are not horizontal or vertical. “Problem 5 of today’s Warm-Up asked if the point (1,5) were on the line y = -5x + 10. “You plugged in the numbers and it satisfied the equation. Any point that satisfies the equation will be on the line. What about (1,6)? Will that be on the line?”
I have them work it out and the general consensus is no, it wouldn’t.
“You have 6 on the left hand side, and 5 on the right. So we could write this as an inequality: 6 > 5. What would you say? Would the point (1, 6) be above or below the line y = -5x + 10?” Most will say above.
“What about (1, 4)? Above or below? Plug it in and solve, and then tell me.” They do so and most will say below. To make sure all are in agreement, we graph it:
“In fact, all points where y>-5x + 10 are going to be above the line. Points where y < -5x + 10 will be below the line. And we can indicate that by shading, just like we did with the horizontal and vertical lines. Let’s do one, and write down the steps.
I provide more examples in this form. Students become used to seeing what side to shade. When they are unsure, I have them do a test point as done above, in which they pick a point on the line and then either increase or decrease the y value of the point to see on which side of the line the point falls.
For inequalities in the form 2x + y > 6, say, students solve for so it is easily graphed. Some examples such as those below. have students solve for y and also determine whether a point satisfies the inequality.
Homework. The homework problems consist of 5 or 6 problems requiring graphing of various inequalities.