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Welcome to Algebra I: Evaluating Expressions

24 minutes

Hey, guys. Welcome to Algebra I. Today we're going to be working on evaluating expressions. To help us get a grip on this, we'll journey back to the order of operations. Ready? Let's go.

(Describer) She uses a stylus on a screen.

All right. Order of operations. Think I skipped one. There we go. When you learned the order of operations, I bet somebody somewhere taught you this acronym: PEMDAS. Remember what it stood for? I bet you do. Let's switch to that pointer tool.

(Describer) P-E-M-D-A-S.

P stood for parentheses, then you got your exponents, multiply, divide, add, subtract. That told you the order that you had to work to simplify an expression. For example--

(Describer) Six minus 2, plus 3 squared.

Let's get the pen back. So, I look at that numeric expression, and I see I've got some parentheses. The order of operations says if I see parentheses, whatever is in the parentheses, I handle that operation first. I've got 6 minus 2. 6 minus 2 is 4. I'm gonna do some mental math, but feel free to use a calculator to make sure that you're right. Or you can do this mental math with me and get your skills stronger. So, 6 minus 2 is 4,

(Describer) She writes.

and I'm not gonna handle that 3-squared yet. Just keep writing that down until you're ready to deal with it. Then order of operations says I would do exponents next. So, I see I have 3-squared. 3-squared is 9, because you remember when you raise a number to a power, you're just multiplying it times itself. So, 3 times 3, that is 9. So, that expression would simplify to 4 plus 9, and I've handled my exponents. I don't have multiplication in my problem. I can check that off. There's nothing to deal with. I don't have division in my problem, so there's nothing to deal with. But I do have some addition. 4 plus 9 is 13. I've handled that, and I'm down to one thing. I'm done. That answer would be 13. I didn't need to subtract. But somewhere along your math career, you handled problems like that using order of operations. Somewhere along the line, PEMDAS changed to GEMDAS, because the math people didn't want you to think that parentheses were your only types of grouping symbols. Because sometimes, like in this problem, you don't have parentheses, but you have an absolute value. Or you may have braces or just anything to group those items together. On this one, I see I have absolute value. I have 9 plus the absolute value of 3 minus 4. Those absolute value bars are a grouping symbol, and I have to handle the operation that's going on in there first.

(Describer) The bars are vertical.

Let's switch to our pen. So, the absolute value of 3 minus 4.

(Describer) She writes.

Absolute value is a way to describe how far away a number is from zero. In this case, first handle that operation inside, and then we're gonna take the absolute value of that number. So, 3 minus 4. That's -1. And the absolute value of -1 is 1. -1 is just one unit away from zero on the number line. A shortcut way to think about absolute value is just whatever your number is inside, make it positive if it's not already. That's a shortcut way. For this one, now that I know that the absolute value of 3 minus 4 is 1, I'll insert that back into my problem. Scoot that over, give myself more room to work. So, the absolute value of 3 minus 4 is 1. So, 9 plus 1. So, I've handled the grouping symbol. I don't have any exponents. I don't have anything I need to multiply. Don't have anything I need to divide, but I do have some addition. So, 9 + 1. That's 10. You're done with this problem. You're down to one number. There's nothing left to subtract. You've simplified that expression: 9 plus the absolute value of 3 minus 4 is 10. Let's keep going.

(Describer) X-squared plus one.

So, here we go. Now, this expression is not numeric. It's algebraic. How you know what the value of this expression is all depends on what number you let take the place of x. For example, let's say I replace x with the number 3. If I let x be 3, let's see what the value of this expression is. I'm gonna replace that x with 3. I'd have 3-squared plus 1. I'd keep going with my order of operations. Don't have any grouping symbols, but I do have an exponent to handle. I've got that 3-squared. That's 9. 9 plus 1. Going down my order of operations. I don't have anything to multiply or divide, but I do have things to add together. 9 plus 1 is 10. If I let x take the value of 3, then that expression has the value of 10. What if I didn't want it to equal 3 and wanted it to equal 4? How would that change the value of this expression? Let's get this out of our way.

(Describer) She erases her writing.

Let's see what happens if x has a value of 4. If I let x be 4,

(Describer) She writes.

then I'd have 4-squared plus 1. Right? All right, so 4-squared. That means 4 times 4. So, that's 16. So, I've got 16 plus 1. That's 17. So, if x equals 4, then that expression equals 17. I've got to set up this pattern for you so you get a handle on this. Let's say x is 5. How does that change the value of this expression? If I let x equal 5. Now what does this expression equal? I'd have 5-squared plus 1. 5-squared. So, that's 5 times 5. So, that's 25 plus 1. 25 plus 1 is 26. Okay. So, depending on what you replace x with, the value of your expression changes. That's how you're gonna be working through this topic on evaluating expressions. The value of the expression depends completely on the value of x. You've got to know the order of operations in order to work your way through these problems.

(Describer) She changes the screen. Titles: Example 1: Evaluate 5x-squared plus 1, when x equals negative 3.

(female describer) She changes the screen. Example 1: Evaluate 5x-squared plus 1, when x equals -3. All right. Let's move along. I'm gonna give you a hint with these. Don't try to memorize every step to what I'm doing. There are a gazillion math problems, and you can't memorize the steps to all of them. You'll be very frustrated. I bet that you've had a moment where you take all the notes in class, you look at your assignment, and you're like, "We didn't do problems like this." You're never gonna see exactly the same problem. It's the process that you've got to master. Don't try to memorize how I did these problems. Memorize my process. Look at, "I'm gonna replace x with some number and then use the order of operations to solve it." Just keep that process in your mind, okay? All right, let's keep going. "Evaluate 5x-squared plus 1 when x equals -3." Okay. So, this problem's telling me that I need to replace x with -3. Let's do that.

(Describer) She writes.

Five. I'm gonna put -3 in the place of x, and I'll put that number in parentheses. Get in the habit of doing that. It helps you get through the problem easier. I see I've got plus 1. Now I'm gonna go to the order of operations, grouping symbol. I see I have parentheses, but there's no operation in the parentheses. It just has that number in there. I can keep going. Exponents. I do see an exponent. I need to go ahead and handle that first. I'm gonna scoot this problem to the left to give me space. Okay. Off to the side, I'm gonna handle that -3 to the second power. If I raise a number to the second power, that means I'm multiplying it by itself. So, -3 to the second power, all that means is -3 times -3. Okay? So, remember your integer operations. When you multiply two negatives together, your answer's always positive. So, a -3 times a -3 is a positive 9. Okay? So, I did that work off to the side. Now I'm gonna get back to my problem. Looks like I left my + 1 hanging over there. Okay. Now that I know... that -3 to the second power is 9, I'm just gonna go ahead and replace that in my problem, 'cause I know that has a value of 9. Okay? I've taken care of the exponent. I'm gonna keep working my way down. Do I see any multiplication? Yes, I do. I have 5 times 9. I'll handle that. That's 45 plus 1. I've taken care of the multiplication. Do I see any division? I don't. Do I see some adding? I do. So, 45 plus 1. That's 46. And you're all done. You've made your way to the end. So, when you evaluate that expression with x equals 3, you get 46. Okay? Let's keep moving. Remember, just replacing that x with whatever value I'm given, and then using the order of operations. All right, Example 2.

(Describer) Evaluate 3 minus 10x, when x equals two-fifths.

(describer) Evaluate 3 minus 10x, when x equals 2/5. Now, I know you saw that fraction, and depending on how comfortable you are, you either cringed or you're like, "I can do it." I hope it was the second one, but if not, it's fine. "Evaluate 3 minus 10x, when x equals 2/5." All right. What do you do first? Remember, just replace x with what you've got. So, if x is 2/5, I'm gonna replace x with 2/5.

(describer) She writes 3 minus 10 times 2/5.

(Describer) She writes 3 minus 10 times two-fifths. Two-fifths is in parentheses.

2/5 is in parentheses. Now, off to the side, I'm gonna handle that multiplication. Let me check off "G." I've handled the grouping symbol. I don't have any exponents. I'm at the multiplication step. Was getting ahead of myself. So, I'm gonna show you a trick of how to handle this multiplying with this fraction. You have 10 times 2/5. I'm gonna use the dot here instead of the parentheses. I'm gonna switch to the dot. Any whole number you can make a fraction by putting it over 1. I'm gonna write 10 as 10/1. So, I really have 10/1 times 2/5. And once you've made your way here, you multiply straight across. I'm gonna scoot that over so you can see all of my work. Okay. I'm going to multiply straight across here. So, 10 times 2. That's 20. 1 times 5. That's 5. Now you just have a simple division problem. 20 divided by 5. Okay. That's 4. What that means is, this part of our problem right here, that 10 times 2/5, it just simplifies to 4. So, I'm just gonna replace that: 3 minus 4. So, that crazy fraction just broke down when you multiplied it by 10, and you got 4. You've handled the multiplication. You don't have any division. You don't have any addition. You do have some subtraction. So, 3 minus 4. That's -1. And you're all done. In that expression, if you let x equal 2/5, your answer's -1. Okay? Let's do a few more. "Evaluate x minus 6 divided by 8, when x equals 13. We have another fraction in this case, but this time our fraction's the expression that we're gonna plug something into. Don't let it get you all bothered. We're gonna use our same steps. I've got x minus 6 divided by 8, and it's telling me to let x equal 13. So, I'm gonna replace that x with 13. So, I have 13 minus 6 divided by 8. All right, let's keep going. I don't have any grouping symbols. I don't have any exponents. I don't have any multiplication. I do have some division. Now, let's kind of back it up a bit. You could think of this top, my numerator in this fraction, as kind of being a group. It may help to think that way, because you have to handle that top part first. I can't start dividing until I simplify that numerator. So, what is 13 minus 6? It's 7. So, I really have 7 divided by 8. And in algebra, we're not afraid of fractions. So, you don't need to divide that any further. Your answer is 7/8. Your answer actually is a fraction. Okay? Let's keep moving. I hope you're getting comfortable with fractions. I'm gonna keep throwing them at you. "Evaluate the absolute value of -7x plus 4, when x equals -2." Okay? So, remember your process. First, go ahead and replace x with that value. So, I have the absolute value of -7 times -2 plus 4. Okay. In this case, those absolute value bars are my grouping symbol. I need to simplify that expression within those absolute value bars first. Okay? So, now, look within the bars. There's a lot going on there. You have to handle the order of operations even within the bars itself. You have a grouping symbol, the parentheses, but there's no operation in there. It's just your number. You don't have exponents, but you do have something you need to multiply together, that -7 times -2. So, off to the side. -7 times -2. Remember, a negative times a negative, that's a positive. So, this is positive 14. I'm gonna bring that into my problem. Let's rewrite this. So, I have the absolute value of 14 plus 4. Okay? Let's keep going. We handled the multiplication. We don't have any division in there. We do have some addition. So, 14 plus 4. That's 18. So, I have the absolute value of 18. We said absolute value, as a shortcut you can think about it as, "Whatever number's in there, make it positive if it isn't already." The absolute value of 18 is 18. That means your answer to your problem is 18. That problem started out large at first. When you let x equal -2, it breaks down to the answer being 18. Okay? I think we've got two more. "Evaluate x plus 4 divided by y minus 3, when x equals -5 and y equals 6." You see I threw you a curveball a little bit. On this one, we have two different variables and two different "replacement values." The numbers you use in place of your variables. are called replacement values, 'cause you're replacing your variable with that value. Don't let it upset you. Just replace them one at a time. We're gonna put them in there. Okay. If x is -5, that means that my numerator, I'd add have -5 plus 4. So, I've got that one in there. Divided by... It said to let y equal 6. So, 6 minus 3. Okay, so I've got that in my problem. On that last one, we said you could think about this as two different groups-- your numerator being a group, and your denominator being a group. You need to handle that first before you get into your division to simplify all of it. Okay. So, -5 plus 4. I'm gonna go across this time with my work. So, -5 plus 4. That's -1. Then, in my denominator, I have 6 minus 3. So, that's 3. You remember we're comfortable with fractions now, right? So, you can just leave that as your answer: -1/3. And you're all done. Okay? Last one we will do together. "Evaluate -4 times the absolute value of x-cubed plus 1," all right, "when x equals 2." We're gonna use our same process. Just start by replacing that x with 2. So, -4 absolute value-- Gonna use parentheses for my 2. 2 to the third plus 1, absolute value. Sometimes people ask, "How do I know if I use parentheses when I replace the value, or do I always have to?" Get in the habit. Whenever you replace x with some value, use parentheses. It won't hurt anything if you do. Sometimes it will hurt something if you don't. Just get in the habit of always using parentheses when you use your replacement values. So, order of operations. The "G" first. I do have a grouping symbol. That absolute value is trapping that group there. I need to handle what's going on in there first. I've got 2 to the third power. So, off to the side. Raising a number to the third power means we're multiplying it by itself three times. So, 2 to the third is 2 times 2 times 2. Okay? Then you just simplify this. So, 2 times 2 is 4, and then 4 times 2 is 8. So, 2 to the third power. That equals 8. Now that I know that this part of my expression simplifies to 8, I'm gonna write that in there. So, I have -4 times the absolute value of 8 plus 1. Okay? We've still got operations going on in that group, so we want to get that worked down. I don't have any multiplication going on in there. Don't have any division, but I do have some addition. So, 8 plus 1. That's 9. So, I have -4 times the absolute value of 9. All right. So, off to the side. The absolute value of 9. Remember? Just make it positive if it's not already. It's 9. Now I know that the absolute value of 9 is 9. Keep in mind that you're multiplying -4 times 9. At this point, I'm putting the dot in there to show that I'm multiplying those numbers together. So, -4 times 9. Remember when you multiply numbers, if your signs are different, then your answer is negative. So, -4 times a positive 9 is -36. And you're all done. You have simplified that expression. All right? Go to the next one. We're at the point where it's your turn to try. Pause the tape. Take your time and work through these problems. Remember, none of the problems are just like any of the ones that we did, because it is the process that you're gonna want to follow to solve these. Remember, replace x with whatever value they gave you, and then use your order of operations to work your way through. After you've worked these out, press play again, and we'll compare our answers.

(Describer) Titles: Evaluate for each given replacement value. Number One: 7 minus 9x, when x equals two-thirds. Number Two: 12 minus 3x, all divded by 5, when x equals 4. Number Three: 2 times the absolute value of x-squared plus y-cubed, when x equals 4 and y equals 1.

(describer) Evaluate for each given replacement value. Number 1: 7 minus 9x, when x equals 2/3. Number 2: 12-3x all divided by 5, when x equals 4. Number 3: 2 times the absolute value of x-squared plus y-cubed, when x equals 4 and y equals 1.

(instructor) Let's see how you did. So, you evaluated 7 minus 9x, when x was 2/3. Let's switch to that pointer tool, move the answer box out of the way. You should have got 1 for that one. "12 minus 3x divided by 5, when x equals 4." On that one, you should have got 0. This last one, you have 2 times the absolute value of x-squared plus y to the third, when x equals 4 and y equals 1. Your answer for that one should have been 34. Okay. All right. I hope you got a handle on evaluating expressions. Remember, replace the value that they gave you for x, and then use the order of operations to work your way through. Hope to see you soon. Bye.

(Describer) Accessibility provided by the US Department of Education.

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In this program, students learn that in an algebraic expression letters can stand for numbers. In algebra, evaluating expressions consists of substituting a specific value for each variable and performing the operations. Part of the "Welcome to Algebra I" series.

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Runtime: 24 minutes

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