Graphing Linear Inequalities with Two Variables:

Linear Inequalities will give us a straight line on a graph. Not only do we need to know how this line looks using the equation, but we also need to understand where the answers to the inequality lies. When we are given inequalities we are usually given a y value that is greater than or equal to another value, that consists of an X variable. We can also be given the greater than/less than or equal to symbol ( ≥ or ≤ ). This greater than or less than helps us decide which side of the line contains the real answers, which we shade in. We also need to determine whether the line is solid or dotted and that is based off if its equal to or not. The solid line means the answers on the line count but if it is dotted they do not. This is easier to understand when we see it in action

Let’s say we are given the equation 4x + 5 ≤ y. We can say that the y-intercept is 5 and that the slope (coefficient of x) is 4/1. This tells us all we need to know to make the line. The line must intercept at 0,5 and it must go up 4 over 1 each time. This is the same form a y = mx+b except it is either greater than or less than and equal to.

To know whether or not which side of the line contains the real answers, we must perform a test. We will insert the point 0,0 into the equation, if it comes out correctly that is the side we shade if it does not, we shade the other side. I will now do this test. 4x + 5 ≤ y, 4(0) + 5 ≤ 0, 5 ≤ 0. Five is not less than or equal to 0, so we will shade the left side. Because the sign includes equal to, the values along the line with be part of it. If it was not equal to the line would not include answers because the answers are less than or more than the line.

That very same test can be used to determine whether or not a point will work in a inequality and if it is a part of the solution, if it does not form a true statement then the point is not part of the solution.

The last thing I will talk about is when we are given a mixed up equation. Let’s say we got the same one as before but arranged differently. 4x ≤ y – 5. We can rearrange this algebraically, all we need to do is add 5 to each side, eliminating it from the y and adding it to the x. Giving us 4x + 5 ≤ y. If y is being multiplied by a negative number and we have to divide it, we need to flip around the sign afterwards, we only have to flip the sign if dividing or multiplying by a negative value.