Inequalities Examples

Graph the inequality 6x + 4y ≤ 1.

First we graph the line 6x + 4y = 1:

Then we pick a point on one side of the line to see whether that point should be included or not. A nice point we can use is (0, 0). If we decide it can't be included, we'll hang a "No (0, 0) Allowed" sign on the clubhouse door and tell everyone else in the treehouse not to acknowledge him.

When x and y are both zero, the left-hand side of the inequality is 2(0) + 4(0), which is indeed less than or equal to 1. That means we do want the side of the line with (0, 0), so we shade in that side of the graph:

Graph the inequality x + 1 > y.

First we graph the line x + 1 = y using a dotted line since we have a strict inequality. No, we don't care if your line is dotted or dashed. As long as it isn't made up of a string of tiny hearts.

Next, we pick a point on one side of the line. Again, (0, 0) is a good one to use. When x and y are both zero, the inequality is totally satisfied, so we shade in the area on the side of the line that includes (0, 0):

Find the inequality graphed below.

First we find the equation of the line, which is y = 4x – 3.

Since we have a dashed line, we know that we want a strict inequality: either < or >. If we take a point (x, y) in the shaded area of the graph, the value of y is less than it would be if we moved up to the line. Apparently, y's currency is worth more above the line, in the "ritzier" part of town.

Therefore, the inequality we want is y < 4x – 3.

To check, we'll try a couple of points. The point (5, 0) is in the shaded area and satisfies the inequality, which is good. The point (0, 0) is not in the shaded area, and does not satisfy the inequality, which is also good. There's more good here than in a box of Good 'n' Plenty.