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Linear programming is a powerful tool that is widely used in business. It is essentially shading inequalities. In your algebra class, you might encounter both one-dimensional and two-dimensional problems. Fortunately, the principles are the same.

## Number Line -- One Inequality

Inequalities have two forms, one that includes the condition of being equal, and one that does not. The inequality x<5 excludes 5, while in x≤5 includes 5. To graph x<5, draw an open circle at 5. This divides the number line into two regions, one below 5 and one above 5. Test the region that includes 0. Is 0 less than 5? Yes. So shade or draw a thick line from the circle at 5 to the left, through 0 and beyond.

## Number Line -- Two Inequalities

Now include the condition x≥-3. Because the inequality includes 3, draw a solid circle at -3 and test. Zero is greater than -3, so shade the region containing 0, to the right of -3. Be sure that you do not shade past the open circle at 5, as you must still meet the condition that x<5.

## Plane Inequalities

In the x-y plane, use dashed and solid lines instead of open or solid circles. Draw a dashed vertical line at x=5 and a solid vertical line at x=-3, and then shade the entire region in between. To shade the two-variable inequality y<-2x + 3, first graph the line y=-2x + 3. Use a dashed line because the inequality is <, not ≤. Then test an x-y point on one side of the line. If the result makes sense, shade that side of the line. If not, shade the other. For instance, (3,4) gives 4<9, which checks out.