okay in this lesson we're going to learn about inequalities so we will be writing linear inequalities sketching graphs of linear inequalities and writing in linear inequalities from graphs so if you see this an inequality is a mathematical sentence that compares expressions an inequality contains the following symbols which i'll explain in a moment to write an inequality look for the following phrases to determine what inequality symbol to use so this is the less than symbol this is the greater than less than or equal to greater than or equal to so the or equal to just has the line underneath um and one way to distinguish the less than from the greater than the uh less than sign is pointing to the left the greater than sign is pointing to the right you might also see um is fewer than is more than and then for the less than or equal to or greater than or equal to you'll see is at most is no more or is at least is no less than and you'll get used to this as you work through some of these problems so for this example we have write each sentence as an inequality so first we'll start with a a number w minus 3.5 is less than or equal to negative 2. so a number w that's just given w so i'm just going to write that down for a w and then minus that's pretty obvious so it's going to write minus 3.5 so is less than or equal to well that is going to be our less than or equal to uh inequality symbol which is this right here so it's pointing to the left and has the line underneath so this means less than or equal to and then negative 2. and this is our inequality based off of the sentence we had for part b we have 3 is less than a number n plus five so i'll write that three and then is less than so that's just going to be less than a number n plus five that should be n plus five and we're done with this one for part c we have zero is greater than or equal to twice a number x plus one so we'll start with zero zero is greater than so or equal to so greater than or equal to so it's going to be this symbol right here it's pointing to the right and it has the line underneath to twice a number x so twice something just means double or times two so two x and then plus one so now we have written our inequalities okay so for these examples we need to tell whether negative four is a solution of each inequality well what i like to say here is what do we do to charge our chromebooks at night which is to plug them in so we need to plug in negative 4. if we get a true statement then negative 4 is a solution and if we get a false statement the negative 4 is not a solution so i'm going to take this negative 4 here and plug it in for x so i'm going to rewrite a as negative 4 plus 8 is less than negative 3. okay well negative 4 plus 8 is going to be positive 4. now i want to look at this and say is this true 4 is less than negative 3. no that's not true 4 is positive negative 3 is obviously negative so this is not true so for a negative 4 is not a solution for part b i'll do the same thing i have negative 4.5 and then i'm going to plug in this negative 4 for x here so i'm plugging with parentheses so i have negative 4.5 times negative 4. is greater than negative 21. scroll down a bit i know that a negative times a negative is going to be a positive and i know that 4.5 times 4 is going to be 18 so it's gonna be positive 18 is greater than negative 21. well this is true okay so this oh go back to part a this is false but this is a true statement and since this is true i know that negative 4 is a solution f so for part b we know that negative 4 is a solution and now we're done the graph of an inequality shows the solution set of an inequality on a number line an open circle that looks like this right here is used when a number is not a solution a closed circle which is filled in is used when the number is a solution an arrow to the left or right shows that the graph continues in that direction so we're going to graph these inequalities i'm going to zoom in a bit so we're going to end up drawing number lines for all of these so here's my number line for part a y is less than or equal to negative three okay so what you want to do is draw your number line and i think it's a good idea to have three numbers on there at least so this is my number line and then i'm gonna put this number in the middle negative three i'll put negative two here and negative four to the left and i want to graph this inequality now so what i'm going to do is i i see that this is a less than or equal to so since this isn't or equal to i want my circle to be filled in i want a closed circle so i'm going to use a different color right here i'll use red so i'm going to put my closed circle on this negative 3 because that's where our value is this means that negative 3 is a solution of this inequality and then since this is less than i'm going to go the numbers lower than negative 3 so that's going to be the numbers pointing to the left so to draw my sketch draw my arrow going to the left okay now you might not have colors available to you if that's the case now always double check with your teacher but if that's the case what i like to do is i like to go like this put it above the number line and then draw it like that okay so this would be the case if you don't want it to be red on because it might look messy without it but either way this is what you can do for part b i'm going to do the same thing i have 2 is less than x well in this case i might want to rewrite it for my x term to be the uh on the left i that's how i personally like to do it but you can do it whatever you want 2 is less than x is the same exact thing as x is greater than 2 right because any number that is that is greater than 2 2 is less than that number so now i'm going to do the same thing i'll draw my number line and i'll draw my three numbers here so 2 is in the middle i'll have 3 and 1 here and now what i want to do is i want to find all the numbers that that x's that are greater than 2. so i'm going to draw a circle at 2 and i'll do this one above this time so draw my circle at 2 and i'm not going to fill it in because 2 is not a solution if 2 is a solution we'd see this the line underneath kind of like in part a but since this is not a solution we want an open circle and then since x is greater than 2 or 2 is less than x we want this arrow going to the right we want all the numbers that are larger than 2 to be in the solution set so i'm going to draw my arrow like that and this is our graph of this inequality now we have x is greater than zero so i'm just going to draw my number line right here and i'll put my three values so i'm going to put negative 1 0 and 1. and once again i do not have a line under here showing me that i have a greater than or equal to this is just greater than so i'm going to put an open circle at zero and then since x is greater than we want all the values larger than zero i'm going to draw my arrow right here and this would be my graph of this number line so this you can do that your graph either way i to be honest find this way less messy but you might see this way as well and i'm going to erase that you might see it like this and you might see it like this either way it's fine so that's how to graph these inequalities for example for the graph shows the height restrictions h in inches for two rides at an amusement park write an inequality that represents the height restriction of each ride so we're going to look at these graphs here so first i'll do ride a i see that i have a closed circle a circle that is filled in so and it's pointing to the right so i'm gonna for right a i'm gonna use h because that's what it tells us to do and then i know that this is gonna be or equal to because it is uh filled in the circle is closed and then my graph is going to the right which means i want all the values 48 and above so i'm going to write this as h is greater than or equal to 48. so this is for part a or ride a and for ride b we can look at it i see that i have an open circle a circle that's not filled in and it's going to the left so in this case i'm going to put h is less than 52 okay so these are my two inequalities once again this where we have greater than because this graph is going to the right here we have less than because it's going to the left now this one it has the or equal to in it because the circle is filled in and this one does not have the oracle 2 because this is an open circle with nothing filled in so that's how to do that one