Solving Linear Inequalities
Linear inequalities are solved basically the same way as linear equations. It can only be solved for one variable in terms of the other. For instance, if we have the inequality 8y minus 4x is less than 8, all we can do is solve it for y in terms of x, or x in terms of y. Let’s solve for y in terms of x. We’ll divide both sides by 8.
That will give us y minus 1/2x is less than 1. Then, we’ll isolate the y by adding 1/2x to both sides. That gives us y is less than 1/2x plus 1. Now we have our inequality essentially in point slope format. You can also graph inequalities in basically the same way as equations. We still have a slope and a y intercept.
We can graph our points here. We know that there is a point right here, because it crosses the y axis at 1, and we know that it has a slope of 1/2, which means that for every one unit it moves to the right it moves half a unit up. For every two units it moves to the right it moves one unit up.
We can plot a second point for this line by moving two units to the right and one unit up. Our second point for the line is going to be up here at 2,2. This first one was at 0,1. These are the two points for the line. Let’s go ahead and draw the line through these points.
This is the line that represents what this would be if it were an equation, but because this is an inequality, y is less than this quantity. To graphically show the solutions for a linear inequality instead of a line, we have a region, in much the same way that with a single variable inequality we had a range of values instead of a single value.
Here we have a region of y that satisfies the equation for every given x. That region, because y is less than this value, is everything where y has a lesser value than this line. Anything below this line, everything down in this region, satisfies this inequality. This is kind of how you can graphically show the solution set for a linear inequality such as this.