In Exercises 103–110, insert either , or = in the shaded area...

Question

In Exercises 103–110, insert either <, >, or = in the shaded area to make a true statement. 30/40−3/4 □ 14/15⋅15/14

Accepted Answer

Welcome back. I'm so glad you're here. We have got a given statement and it's two absolute values and we're trying to figure out if they're less than greater than or equal to each other. So those a little bit bigger. We've got the absolute value of eight squared times negative one cubed divided by four plus 10. And just note that I'm writing divided by with the symbol and there you see it with the slash. Those are the same thing and then end the absolute value. And now this is exactly what we're looking for. That's what that box means. It means fill in the blank box. So is that greater than less than or equal to the absolute value of five cubed divided by parentheses, three squared plus four squared and parentheses and absolute value. Great. Now how do we go about solving this? Well we can recall from previous lessons that we've got this handy acronym penned us to remind us of the order of operations, the order in which we do all the various operations taking place and we can start on the left hand side and just do the left hand side all the way down first until we get an answer and then come back and do the right hand side. We don't have to jump back and forth between them looking for the operations in that order. So we'll start on the left hand side and we'll look for parentheses. Are there any parentheses there? Well you can see this parentheses around the -1. But there are no operations happening inside those parentheses. So those parentheses are actually just indicating that it's this whole number in here negative one that's getting cubed. Um And so you wouldn't think that the negatives on the outside and that it's just the one getting cubed. They want to make sure you know it's negative one that's being cubed. So that's the function of those parentheses and it doesn't count for our pending order of operations. Now we look at exponents. Well yep there are plenty of exponents so let's just work from left to right. Eight squared eight squared is 64. We've done that operation and now we can just bring the rest down absolute value 64 times negative one cubed divided by four plus 10 and absolute value anymore. Exponents. Yes this negative one cubed and we know that negative one cubed is negative one and we can bring down the rest absolute value of 64 times negative one divided by four plus 10. And we have got our exponents done there's no more. So now we'll look at multiplication and division. What does this mean? It means that we start on the left and do the first multiplication or division that we see going in order from left to right. So 64 times negative one is the first multiplication or division that we see we can go ahead and do that 64 times negative one is negative 64. Bring down the rest absolute value of negative 64 divided by four plus 10. And now anymore multiplication or division yep negative 64 divided by four that is negative 16. Bring down the rest, absolute value of negative 16 plus 10 And now multiplication and division are done. All that's left is addition and subtraction and that's all that's going on. There is negative 16 plus 10 and so we can do this negative 16 plus 10 is negative six. And it's still bringing down the absolute value absolute value functions. A little bit like parentheses. It's a little bit like we've done everything inside the parentheses here and so you could say now we're clearing that last bit. So what is the absolute value of negative six meaning how far is negative six from zero on the number line, -6 is six units away from zero on the number line. So that is the absolute value of -6. Now let's work on the right hand side, starting at the beginning of pandas. Again, do we have any parentheses? Yes we do. We've got parentheses but it has several different operations happening inside so we're not going to clear the parentheses yet. We have to do those operations all first. So what's the first operation inside of the parentheses? Well there isn't another parentheses inside the parentheses but there are exponents. So let's work on those. We've got three squared three squared is nine. We can bring down the rest absolute value of five cubed divided by parentheses nine plus four squared. And now we've got another exponents inside that parenthesis and we'll do that one. four squared is 16. Now we can bring down the rest absolute value of five cubed divided by parentheses nine plus 16 and parentheses. And now the only operation in those parentheses is addition. So we can do the addition and clear out those parentheses. Nine plus 16 is 25. Bring down the rest absolute value of five cubed divided by 25. Now there's no parentheses that one's done and we can do exponents. We've got five cubed. Five cubed is 125. We can bring down the rest 125 divided by 25. And that's the only operation left inside the absolute value bars. So 125 divided by 25 is five but it's still the absolute value of five, meaning how far is five from zero on the number line? Well five is five units away from zero on the number line. So we've got six and five and now we can assign which one's greater. Six is greater than five so that is the answer we're looking for which matches with answer choice. A well done. We'll catch you on the next one

In Exercises 103–110, insert either <, >, or = in the shaded area to make a true statement. 30/40−3/4 □ 14/15⋅15/14

Key Concepts

Fractions and Operations

Equivalent Fractions

Order of Operations

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