Algebra 1 Basics for Beginners

Welcome to another video from UltimateAlgebra.com. In this video, we will be looking at how to answer Algebra 1 questions the easiest way. Please, this video will not be exhaustive. For a complete course, deeper dive, with more examples, please check out our Ultimate Algebra course at UltimateAlgebra.com. Let's dive right in. Question 1. x plus 2 equals 5. Solve for x. The first thing we are looking at is how to solve one-step equations. You have to be good at solving one-step equations in order to pass any math test. The idea of solving equations is to move everything with the x to the other side of the equation by performing the opposite operation on it. We want only the x to be on one side of the equation. So here, we want to move the plus 2 to the other side of the equation. We do that by performing the opposite operation on both sides of the equation. The opposite operation of addition is subtraction. The opposite operation of multiplication is division. The opposite operation of exponent is the root or radical. We know the opposite operation of addition is subtraction, so we will subtract 2 from both sides. Once the opposite operation is done, those two numbers simply cancels out. So here, the 2 will cancel out. 5 minus 2 is 3, so x will be equal to 3. We have a free complete video on solving equations with a lot more examples. Please check it out with the link in the description for more. Let's move on to question 2. Question 2. 2x plus 3 equals 11. Solve for x. Here we are looking at solving two-step equations. We said earlier that the whole idea of solving equations is to get rid of everything and leave the x on one side of the equation. For this question, we will see that we have to get rid of the multiplication by 2 and the plus 3. 2x is the same as 2 times x. When there are more than one operations, we use the idea of the reversal of the order of operations to know which one to get rid of first. So here is the order of operations. If you are not familiar with the order of operations, please revise our videos on it. It's very important. Knowing the order of operations will make solving of equations super easy. We can see that in the reversal of the order of operations, that's from bottom to top, we have addition before multiplication. So we will get rid of the plus 3 first. We get rid of the plus 3 by performing the opposite operation on it. So we will subtract 3 from both sides. The 3 will cancel out. 11 minus 3 is 8. So we have 2x equals 8. Next, we will get rid of the multiplication by 2 by dividing both sides by 2, since division is the opposite of multiplication. The 2 will cancel out. 8 divided by 2 is 4. Therefore, x equals 4. Question 3. 3x squared plus 8 equals 20. Solve for x. We are looking at solving multi-step equations. The process of solving is just like solving two-step equations. We want to get the x on one side of the equation. In order to do that, we have to get rid of the multiplication by 3, the exponent 2, and the plus 8. We will use the reversal of the order of operation to know which one to perform first. Let's bring our order of operations. So here is our order of operations. In the reversal, we will notice that we have to do the plus 8 first. Then we will do the multiplication by 3, and then we do the exponent 2. We will get rid of the plus 8 by performing the opposite operation on it. Subtract 8 from both sides. The 8 will cancel out. 20 minus 8 is 12. So now we have 3x squared equals 12. Next, we have to get rid of the multiplication by 3. We do the opposite operation. We divide both sides by 3. The 3 will cancel out. 12 divided by 3 is 4. So now we have x squared equals 4. We finally have to get rid of the exponent 2. We do the opposite operation. The opposite operation of squared is square root. We find the square root of both sides. This will cancel out. The square root of 4 is 2. Therefore, x equals 2. Question 4. 4x plus 5 equals 9 plus 2x. Solve for x. In our previous questions, we have been having the x represented only once. Example. 4x plus 5 equals 9. But here we have the x represented twice. In a case like this, we want to move the x values to one side of the equation and work on it. So here, you can choose to move the 4x or 2x. I'll move the 2x to the other side of the equation. To do that, since the 2x is adding, you will subtract 2x from both sides of the equation. The 2x will cancel out. Now 4x minus 2x is 2x. So we have 2x plus 5 equals 9. We now have a familiar two-step equation, which you should be able to solve if you've watched the previous questions. But let's go over it. We want to get rid of all the numbers attached to the x, so we can have the x alone on one side of the equation. To achieve this, we know that we must get rid of the times 2 and the plus 5. Let's bring our order of operations. Please, you don't need to be writing the order of operations in your solutions. We are using it for teaching purpose. We are using the reversal of the order of operations, so we are working from the bottom up. We will see that in this form, we must do the addition first before the multiplication. To get rid of the plus 5, we must perform the opposite operation on both sides of the equations. So we will subtract 5 from both sides. The 5 cancels out. 9 minus 5 is 4. We now have 2x equals 4. Next, we will get rid of the multiplication by 2. We do this by performing the opposite operation on both sides of the equation. So we will divide both sides by 2. The 2 will cancel out. 4 divided by 2 is 2. Therefore, x equals 2. Question 5 The absolute value of x plus 3 equals 7. Find x. For absolute value equations, we equate the absolute value to the positive and negative of what is on the other side of the equation. Here, we will equate the absolute value to positive and negative of the 7. So we have x plus 3 equals 7, and x plus 3 equals negative 7. We solve both equations. For the first one, we subtract 3 from both sides. The 3 will cancel out. 7 minus 3 is 4. For the second one, we subtract 3 from both sides. The 3 will cancel out. Negative 7 minus 3 is negative 10. So our answer is x equals 7 and x equals negative 10. Question 6 The absolute value of x plus 1 plus 6 equals 9. Find x. In the previous question, everything on the left side was in the absolute value marks. Here, the trick is that the plus 6 is not in the absolute value. You need to remove everything that is not in the absolute value mark and get only the absolute value on one side before you can equate to the negative and positive. So we will first start by subtracting 6 from both sides. The 6 will cancel out. 9 minus 6 is 3. Now we have the absolute value of x plus 1 equals 3. We have only the absolute value on one side of the equation. So we can equate x plus 1 to the positive and negative of what is here as usual. We have x plus 1 equals negative 3 and x plus 1 equals positive 3. We subtract 1 from all sides. The 1 will cancel out. For this one, negative 3 minus 1 will be negative 4. So x equals negative 4. For this one, 3 minus 1 will be 2. So x equals 2. So x equals negative 4 or x equals 2. Question 7. Square root of x plus 3 minus 2 equals 1. Find x. We are looking at radical equations. A radical equation is an equation in which the variable is contained inside a radical or root sign. Here, we see that the x is under the square root sign. Similar to what we did for absolute value equations, we want to isolate the radical on one side of the equations. We add 2 to both sides. The 2 will cancel out. 1 plus 2 will give us 3. Now we have the square root of x plus 3 equals 3. Next, we want to eliminate the square root by squaring both sides of the equation. For the left side, the square root will cancel out the square. We just get x plus 3. 3 squared is 9. We have a one-step equation. We subtract 3 from both sides. The 3 will cancel out. 9 minus 3 is 6. Therefore, x equals 6. Question 8. 4 divided by the quantity x minus 5 equals 3 divided by x. Find x. We are looking at rational equations. For rational equations, we have the x or variable in the denominator. The first step is to remove the fractions. Typically, we will use the least common denominator method, but for this question, we can just cross-multiply. 4 times x is 4x. 3 times x minus 5 is 3x minus 15. We expanded it. 3 times x is 3x and 3 times negative 5 is negative 15. Next, we want to isolate the x on one side of the equation. We subtract 3x from both sides. The 3x will cancel out. 4x minus 3x is 1x or simply x. So we have x equals negative 15 as our answer. Question 9. Solve for x, given that y equals mx plus b. Here, we are looking at change of subject or transposing of formula. This is the slope-intercept form of the equation of a line. When you are given any formulae, you should be able to find any of the variables. This is nothing different from what we've been doing so far. To solve for x, we want to have the x on one side of the equation and everything else on the other side of the equation. To do that, we have to move the times m and the plus b. We will use the reversal of the order of operation to know which one to perform first. Let's bring our order of operations. In the reversal, we will notice that we have to do the plus b first. Then we will do the multiplication by m. We will get rid of the plus b by performing the opposite operation on it. Subtract b from both sides. The b will cancel out. You cannot subtract y minus b because they are dissimilar as we learned in addition and subtractions in algebra. So we will have y minus b equals mx. Next, we have to get rid of the multiplication by m. We do the opposite operation. We divide both sides by m. The m will cancel out. There's nothing we can reduce on this other side. So x equals y minus b all over m. Question 10. Negative 3x plus 1 is greater than 7. Solve for x. Here, we are looking at solving inequalities. The process is the same as solving equations. There is a slight difference when you multiply or divide by a negative. We want to get rid of everything and leave the x on one side of the inequality sign. For this question, we will see that we have to get rid of the multiplication by negative 3 and the plus 1. Let's bring our order of operations. We can see that in the reversal of the order of operations, that's from bottom to top. We have addition before multiplication. So we will get rid of the plus first. We get rid of the plus 1 by performing the opposite operation on it. So we will subtract 1 from both sides. The 1 will cancel out. 7 minus 1 is 6. So we have negative 3x is greater than 6. Next, we have to divide both sides by the negative 3 so we can get the x by itself. In inequalities, when you divide or multiply by negative, the inequality changes. So here, the greater than becomes less than. Please take note of this. Nearly all wrong answers are because of this mistake. Now the negative 3 will cancel out. 6 divided by negative 3 is negative 2. Therefore x is less than negative 2. Question 11. Solve the inequality, negative 3, less than x plus 8, less than 20. Here, we are looking at combined inequalities. This question is the same as negative 3 less than x plus 8, and x plus 8 less than 20. We just combine them. The solution is exactly the same. Instead of having two sides, you now have three sides. To get the x by itself, we have to subtract 8 from all three sides. The 8 will cancel out here. Negative 3 minus 8 is negative 11. Then 20 minus 8 is 12. So our answer is negative 11, less than x, less than 12. Question 12. Graph the inequality x greater than negative 4. Here, we are looking at graphing inequalities. Let's bring our number line. When graphing inequalities, the first thing is to locate your point, which will be the number. Here, it is negative 4. Then you'll draw a shaded or unshaded circle at the point. If you have less than, or greater than, then the circle will not be shaded. If you have less than or equal to, or greater than or equal to, you will use the shaded circle. So basically if it has an equal to, you will shade. In this case, since it's just greater than, we will not shade the circle. Then finally, we draw an arrow. The easiest way to get this always right without thinking is to make sure your inequality is in the form variable, inequality sign, and then the number. In that form, the direction of your arrow will be the same as the direction of the inequality sign. Here, since this is in that form, we will draw our arrow facing here, and we are done. Question 13. In order to ship 2,500 gallons of a product to another country, Stan's shipping company had to package the gallons into boxes. If they had 20 packaged boxes and 100 gallons left that are not in boxes, how many gallons were in a box? This kind of two-step equation word problem is very common. We are first going to solve it in details for teaching purpose. Then I'll show you how you can solve it in less than 10 seconds on an actual test. You have three values in these type of questions. Let's write them down. We have 2,500 gallons. We have 20 boxes. And finally, we have 100 gallons. The 2,500 gallons represents the total. So we have equals 2,500. The 20 boxes is what I call the group. The gallons have been grouped into boxes. For most questions, the group can also be identified as the number that represents something different from the other two numbers. So here, 2,500 represents gallons. The 100 also represent gallons. But the 20 represent boxes. So the 20 will be the group. The group is the one with the x, so we will have 20x. We can now add the 100 gallons left to the equation and solve for the x in this two-step equation. Subtract 100 from both sides. These will cancel out. 2,500 minus 100 will be 2,400. We now have 20x equals 2,400. Divide both sides by 20. The 20 will cancel out. 2,400 divided by 20 will be 120. This means there were 120 gallons in each box. The hard part of this question is to be able to pull out the values from the word problem. We went through a detailed solution for teaching purpose. Let's look at how you can speed up solving this question. First, there's absolutely no reason to write this part if you know what you're doing. You can go straight to writing your two-step equations. We have our group, 20x. Our other, plus 100. And our total, 2,500. Then you can solve the two-step equation. An even faster method to solve this question in less than 10 seconds is to do 2,500 minus 100, divided by 20 on your calculator to get 120. We did the total minus other, divided by the group. A big caution when using fast methods is to note that they are very specific to specific questions, and little twists to the question can let you get it wrong. So when in doubt, use the longer methods. Question 14. 5 added to thrice Michael's age is 50. How old is Michael? For a question like this, it is good practice to start with identifying your unknown value and representing it by a letter. Let's say X. The unknown value is usually what the question is asking you to find. Here, it is Michael's age. Let's represent it with X. Now we just translate. We know added to is addition. Thrice is three times. So thrice Michael's age is 3X. And is means equal to. So this is 5 plus 3X equals 50. You can now solve the two-step equation. We subtract 5 from both sides. The 5 will cancel out. 50 minus 5 will be 4 to 5. So we have 3X equals 4 to 5. Next, we will divide both sides by 3. This will cancel out. 4 to 5 divided by 3 is 15. Therefore, X equals 15. Michael's age is 15 years. Question 15. Which of the relations below is not a function? To answer this question, let's go through a few things about functions. When you have a notation, example, 2, 5, the 2 represents the input value. That's X value. The 5 represents the output value. That's the Y value. This is the same as what we will learn in graphing points. For a relation to be a function, no input value can have multiple output values. Also, all input values must have output values. Let's look at examples. Here, the input values are these, and the output values are these. Note that each input value corresponds to only one output value. Although 3 is an output value without any input value corresponding to it, it is still a function. This could have been written in this form. We wrote each of the input values and what output value they correspond to. Now let's look at this relations. This relation is not a function because the input 2 has more than one output value. That's 7 and 5. No input value can have more than one output value. This could have been written this way. Notice we have two written twice with different Y values. One is 7, and the other is 5. Finally, let's look at this relations. This is a function. Multiple input values can go to the same output value. So here, although the input value 4 and the input value 5 both gives 8, it is a function. Again, this could have been written this way. Notice we have our Y value 8 written twice. With this information, let's answer our question. The answer is C. C is not a function. The input value 3 has two output values. That's 6 and 8. For a function, no input value can have more than one output values. Happy Thanksgiving! Get the full course at ultimatealgebra.com and change your algebra forever! Watch more of our videos on YouTube. Please like, share, and subscribe!