Y is greater than or equal to 2x 3
The coordinates would be (0,3)
The slope would be 2/1 so you go up 2 and over 1 but is it left or right?
When you graph it it will have a dark line. Which way would you shade it left or right?

Good job so far.

First, the tried and true method: When graphing inequalities, once you have the function plotted (a line in this case), all you need to do is pick a test point on one side of the line and see if it makes the inequality true or not. If it's true, then shade in the side that you chose. If not, shade in the opposite side (but you might want to pick a point on that side and verify that the inequality is indeed true to make sure you didn't make a mistake).

For example, in this case if you pick the origin, (0,0), that happens to land to the right of the line. Plug in the x- and y-coordinates of that point into the inequality:

y \ge 2x + 3

0 \ge 2 \cdot 0 + 3 ?

0 \ge 3 ?

Clearly 0 is NOT greater than or equal to 3, so it's the opposite side (the left) that should be shaded.


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The above method works every time, but you can skip all of the plotting test points nonsense if you just take a minute to understand what the inequality represents.

First let's pretend it was an equation rather than an inequality (in other words pretend it had an equals sign instead of a greater than or equal). Like any function, it represents all of the possible points in the universe with x- and y-coordinates that make the equation true. In this case it's the set of all points whose y-coordinate is 3 more than twice the x-coordinate. There are an infinite number of them, but if you were to put a dot on a graph for a whole bunch of them (so many dots that they got so close together that the ink ran together between them) you'd see that they formed a line. Luckily, there are shortcuts to plotting lines so you don't need to plot thousands of dots on a graph. You happened to use the slope/intercept shortcut, and from what you described, you plotted the line just perfectly. :) Regardless, if you were to pick any point on that line (and remember it extends infinitely in both directions), you'd find that it satisfied the equation y=2x+3.

Now what about the case of the inequality? Here the function represents not only those points where y EQUALS x+3, but also those points where the y-coordinate is GREATER than x+3. So where's the region of the graph where the y-coordinates are greater than the line you plotted? It's ABOVE the line, of course! Any point above the line has a y-coordinate greater than those points on the line.

Since this is a line which goes up and to the right (or down and to the left depending which way you want to think about it), points above the line are the same as points to the left of the line, so the "above the line" answer using this method agrees with the "left of the line" answer we got above using test-points. But when you asked if you should be shading the region to the left or right of the line, what you really should have been asking is if you should shade the region above or below the line.

Either way, if you've managed to read all the way through this long-winded answer, hopefully you've got it by now!

Wow thank you I wish my teacher explained it like this it would've made the problem easier.

Thanks for the feedback! It's always nice to hear that I helped make something a little clearer. Math teachers (which I'm NOT) sometimes have a way of making simple things seem excessively complicated. ;)

You're welcome. I thought you were a teacher but guess you're just really smart. Thanks again!