[Music] what's up everyone and welcome to this in-depth lesson on graphing linear inequalities now at this point we already have a whole collection of algebra skills that we're going to apply to this New Concept let's just start off with solving a simple equation in this case x + 5 is equal to 8 which we know has a solution of x = 3 since 3 + 5 is equal to 8 now if we think of the solution to this simple equation in terms of a number line we know that three is the only possible solution that any other value would not work for X since there's no other number that you could add five to that would equal eight so this simple linear equation only has one possible solution however What If instead of an equal sign this was an inequality so now we have x + 5 is greater than or equal to 8 if we were to solve this our solution would be that X is greater than or equal to three which means that X can be three or any value that's larger than three so now our solution includes the number three so we're going to use a closed Circle since it's greater than or equal to and anything that's greater than three so we have to shade to the right so we can say that three is a solution and any value that is larger than three is a solution and we can also say that any value that is less than three is not a solution and since we know that there are an infinite amount of numbers that are larger than three we can say that the solution to an inequality is a set of an infinite number of solutions now before we move on let's imagine that that inequality sign was a greater than symbol instead of a greater than or equal to symbol in this case our new solution would be X is greater than three which means that 3 is no longer included so our circle is not shaded anymore and now we can say that our solution set is any value that's larger than three but not including three so now that we've refreshed all our prior knowledge of equations and inequalities let's ask ourselves what are inequalities going to look like on the coordinate plane now we already know that in linear equations such as Y = X or y = x that any point on the line is going to be a solution to those equations but we're interested in inequalities here so if these were inequalities instead of equations they would look a lot different if we had Y is greater than or equal to X we would have to shade all the area above the line and if we had Y is less than or equal tox we'd have to shade all the area below the line now this shading is dependent on the inequality sign when we have a greater than or equal sign we are going to shade above the line and when we have a less than or equal sign we're going to shade below the line so maybe now your mind is blown and you're asking what does this mean we already established that when it's an equation all of the points on the line are solutions now with inequalities not only those points on the line are solutions but the points in the Shaded area are all solutions as well and later on in this lesson we'll explore why now this is the case when we have the greater than or equal to or less than or equal to symbol if we were to replace those inequality symbols with just greater than or less than like we're going to do right now this changes things a little bit because now the points on the line are no longer included in the solution and only the points in the Shaded area are in these cases we'll be using a dash line to indicate that the points on the line are not included in the solution so if you're seeing this for the first time it might be a little confusing so before we get into any examples let's just try to clarify here if we add the inequality X is greater than or equal to two we could say that the solution is any value greater than or equal to two so any number larger than two and including two however if it was just a greater than symbol X is greater than than two our solution will be any value greater than two but would not include two so when graphing these linear inequalities the greater than or equal to or less than or equal to symbols have a solid line where the points on the line are included in the solution and the greater than or less than symbols have a dash line where the points on the line are not included in the solution so now we are ready to check out our first example and we'll consider the linear inequality Y is greater than or equal to 3x - 2 we can graph this just like a linear equation by starting at the Y intercept -2 and using the slope to build the line and since our symbol is greater than or equal to our line is going to be solid that greater than or equal to symbol also means that we have to shade above the line so the entire region above the line is going to get shaded in so now we have the completed graph of this linear inequality and now we can go ahead and choose points to see whether or not they would be a solution so let's start by checking a point in the Shaded region so we're going to choose the point with coordinates -55 now those are XY coordinates so we can replace those two values the x value with5 and the Y value with positive 5 5 and now we just have to evaluate so 3 * -5 = -15 and -15 - 2 = -7 so we're left with 5 is greater than or equal to -17 which is true and should help us to understand why any point in the Shaded region will be a solution to this linear inequality next we'll look at a point that is on the line in this case the point with coordinates 2 4 so again we're just going to replace the x value with two and the Y value with 4 and then we can evaluate so we know that 3 * 2 just equals 6 and 6 - 2 equals 4 and what we are left with here is four is greater than or equal to 4 which we know is true and this should help us to understand why any point on the line is also included in the solution set now finally what about a point that's in the non-shaded region let's choose the point with coordinates 60 so again we'll replace the x value with six the value of our x coordinate and the Y value with Zer the value of our y-coordinate for this particular point in the non-shaded region now 3 * 6 is equal to 18 and 18 - 2 is equal to 16 so now we're left with 0 is greater than or equal 16 that of course is not true and this should help us to understand why points in the non-shaded region are not included in the solution so to recap any point in the Shaded region or on the line is a solution and any point in the non-shaded region is not so now we're going to look at one more example in this case we'll be examining the linear inequality Y is less than - 12x + 4 so again we're going to graph this line and since it's a less than symbol we are going to use a dash line instead of a solid line and less than means that we need to shade the region below the [Music] line so now we're ready to check out some points and we'll start again in the Shaded region and let's use the origin 0 0 it's a really easy point to use so again that's an XY coordinate so I just have to replace the x value with zero and also the Y value at zero because it's the same value for both X and Y and then I can just evaluate - 12 * 0 is just 0 and 0 + 4 is equal to 4 so we're left with 0 is less than four which we know is true and this should help us understand again why any point in the Shaded region will be a solution to this linear inequality next we'll look at a point that is on the line and let's choose the0 04 four okay so now again we want to replace the x value with zero and replace the Y value with four and then just evaluate and then we can check and see if it's true or not so again - one2 * 0 is just 0 and 0 + 4 just equals four now we're left with four is less than four that is not true so this should help us to understand why when we have a dash line at the points on the line are not included in the solution set and finally let's see why by a point in the non-shaded region will not be a solution so let's go with the 68 so again replace the x value with six and the Y value with eight and evaluate so - one2 * 6 is equal to -3 and -3 + 4 is equal to POS 1 leaving us with 8 is less than one which we know is not true and this should help us understand why any point in the non-shaded region is not a solution so now we can better understand why that when we have a dash line only the points in the Shaded region are in the solution set and all the points on the line and in the non-shaded region are not and that is it for this lesson we finished another one thanks again for stopping by see you thank you again everyone for joining us and please reach out to us on Twitter @ mashup maath we are dying to hear from you so please share some love all right we're done here