 ###### Devorah Goldblatt

Case Western Univ., summa cum laude
Perfect scorer on the SAT & the ACT

Devorah is the founder of Advantage Point Test Prep and the author of the book “Boost Your Score” The Unofficial Guide to the Real ACT.

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# Math Content Review II

Devorah Goldblatt ###### Devorah Goldblatt

Case Western Univ., summa cum laude
Perfect scorer on the SAT & the ACT

Devorah is the founder of Advantage Point Test Prep and the author of the book “Boost Your Score” The Unofficial Guide to the Real ACT.

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Let's get started talking about the top elementary algebra concepts you'll see on the Math ACT. We're going to talk about substituting the variable, simplifying equations, solving linear equations, and inequalities. And if you're thinking, "Oh my gosh, that sounds really overwhelming," Don't worry we're going to break it down.
Let's start talking about substituting the variables. These questions give you an algebraic equation and then just the value of a variable within and they're going to plug it in. These are really straight forward. Let's look at this one, it looks complicated when you first look at it, but take a close look, when x plus y is 3, what is the value of two times parenthesis x plus y plus x plus y over three minus parenthesis x plus y to the fourth. So again it looks horrible but what do we know here, we know x plus y is equal to three and now this is the case where you'll just plug that in wherever you see x plus y. So it turns into a really straight forward problem, we have two times three, like x plus y, plus three over three or just one, minus three to the fourth, okay great. So just to simplify, this is six plus one minus 81, okay, and that is equal to six, seven minus 81 or negative 74, and we've got it. So it wasn't that bad after all, the answer choice A would be correct.
Let's keep going. Let's take a look at simplifying equations, so x squared minus four x plus nine in parenthesis and in another parenthesis, we have three x squared minus two x minus six. It looks kind of complicated you're thinking, "Oh my gosh, is there multiplication going on, what's up with the parenthesis." But if you take a close look, all we doing is subtracting, so these question types are just testing, can you combine expressions. These you remember from algebra, you know in school, combining like terms. So let's go ahead and get started with this problem. So we're going to get rid of the parenthesis, x minus four x plus nine, and now the next part, students often mess up on this so keep an eye out. You want to distribute your negative, it is so important it needs to go to every one of your terms, because your parenthesis means that, it's got to travel to every one of the parts. So minus three x squared, plus two x 'cause you've got two negatives there, so plus two x and then also plus six, great. And now this is looking pretty straight forward right? And now you know you just have to combine like terms, which you probably remember doing you know from class. So we've got x squared minus three x squared, okay just negative two x squared, so let's cross this out this becomes negative two x squared. Next you've got your negative four x plus two x, so negative four x plus two x just negative two x, and I can cross these out too. Okay last, nine plus six just 15, and we've got it. We've just simplified these expressions perfect. So which of these choices matches this? We need negative two x squared minus two x plus 15, and C is the correct answer choice here.
Let's move on, next solving linear equations. Again pretty straight forward but it's worth reviewing. If five x plus three is 23, what's 12 x minus ten? So the linear equation are just those really straight forward ones, you know we're not talking x squared, x cubed, just a plain x, often just one variable and that's what you're going to do, you're going to figure out what that x is? Five x plus three is 23 and then there is another step, okay, so what's 12 x minus ten. So you just take it into parts, if five x plus 3 is 23, first of all what's x? So five x plus three equals 23, so you know you need to isolate the variable, we're going to take three to the other side, okay so we have that five x is equal to 20. And in that case x is going to be equal to four, right? Cause you divide both sides by five. Okay, but we not done, and by the way you guys a tip for you, once in a while the ACT people are kind of mean that actually one of the answer choices would be, the answer that you'll get the first half of the problem, if it's a two part problem like this. So keep an eye out, here they're nice, we don't have a four, but sometimes there might be. Okay we know we're not done, we need to plug in our four to our x, back into 12 x minus ten, let's do that, so 12 times four minus ten, okay we know we do our multiplication before a subtraction, 48 minus ten which is going to equal 38. So B is the correct answer choice here. And you see how even though this looked you know, a couple of steps maybe a little tricky, it's really pretty straight forward if you remember the stuff you learn in school about isolating your variable.
Let's keep going and look at Inequalities. These are one of my favorite topics on algebra. An inequality states that one side of the equation is greater than the other. So just review of what the sign means, if you have A, this is A is less than B, if you've got, my teacher I think it was fifth grade, think about it you probably had this to, she says that the crocodile eats the bigger side you know, so if you had an open mouth, it would be pointing to the larger part of the equation, that's how I remember it, here because the mouth is pointing to the B, the B is larger, so A is less than B, A is greater than B, A is less than or equal to B if you've got this little line here, And A is greater than or equal to B, okay just a quick review.
Let's do a problem, now let me tell you something important, these look kind of intimidating but actually do them how you would do a straight forward algebra problem, pretending that this was actually an equal sign. The only time anything is ever going to change is if you multiply or divide by a negative number, watched out for this, they test this on the ACT and in that case, you're going to flip your inequality. Okay, so if four times x minus three minus two, is greater, okay let me write this out, four times x minus three minus two is greater than or equal to three times four x plus five. Okay, let's do it how we would do a typical equation, and again you're going to what to isolate the variables so we move some stuff around. First though, you know you need tackle out the parenthesis, so we've got to do four x minus 12 here and distribute four x minus 12 minus two is greater than or equal to 12 x plus 15. Okay this is already looking simpler, now you know this is end, this ends the thing, four x minus 14, 14 greater than or equal to 12 x plus 15, now we can move the 14 to the other sign and then we can move the 12 x to the side. Let's do that at the same time, so we have the 12 x moving over here, so you would have, negative eight x and that if you move the 14 to this side and you add it, you going to have its greater than or equal to 29. Okay we're almost finished, now all we have to do is divide by negative eight on both sides. And yes I said negative, we're dividing by a negative number, so we're going to switch the inequality and it's going to look like this, divide by negative eight, divide by negative eight. So now x is less than or equal, I flipped it, than 29 over negative eight. And actually you can't simplify that any further because 29 is prime. Great, which answer choice looks like that, and we see it's answer choice D, x is less than or equal to 29 over negative eight. And be careful because always one of the answer choices will be the same answer if you didn't flip the inequality, okay.
That's it for the Top Elementary Algebra Topics you'll see on the ACT. And again there are ten of these that you will see in the Math section. Now if you still feel like you need a little bit of work that's no problem, head to the bonus material and there you will find a lot of resources with extra practice problems.
On to intermediate algebra. These are the two concepts you'll see showing up most frequently for intermediate algebra on the ACT, and remember there are nine intermediate algebra questions. First solving quadratic equations and then solving systems of equations we'll go over this. Let's start with the first one quadratic equations. Hopefully you guys remember what these are from Math class. This an equation where something is squared and they're usually in the form AX squared plus BX plus C and this when you factor them with your two sets of parenthesis and you find the solutions. So let me tell you something really neat actually your calculator has a really good program that will calculate this for you. Students don't know that programs on your TI calculator are okay on the test but they are. So check your bonus material and I actually have a program there that will do this equation for you all you have to do is put in the number that shows up here in each of them. So here there's no number you put one you'd put four, you'd put four and you would do tup tup turup, take a second calculate your roots no brain power required on your end.