WEBVTT mathematics/algebra-2/eaton
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Welcome to Educator.com.
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In today's lesson, we are going to be covering solving absolute value equations.
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Recall that the absolute value x, written as |x| (this is the symbol for absolute value), is the distance from x to 0 on a number line.
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For example, if you have the absolute value of x equals three, what you are saying is
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that the absolute value of this number is 3 away from 0 on the number line (1, 2, 3).
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OK, so looking at this: if I go from 0 to 3, this is 3 units away from 0 on the number line; therefore, x could equal 3,
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because 3 is 3 away from 0 on the number line.
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However, consider this: -3 is also 3 units away from 0 on the number line.
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So, x could also equal -3; so, if the absolute value of x equals 3, x could be 3 (since the absolute value of 3 is 3),
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and x could be -3 (since the absolute value of -3 is also 3).
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Knowing this, and really understanding this definition, will allow you to solve equations involving absolute value.
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Again, absolute value equations: you can solve equations containing absolute values, using the definition of absolute value.
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For example, |9x + 2| = 29; well, I already said that, if the absolute value of x is 3, that means that x equals 3, or x equals -3.
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So, you can apply that same concept right here: 9x + 2 = 29, or 9x + 2 = -29.
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So, once you remove the absolute value bars (and you can do that by turning it into two related equations,
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one where it equals the positive, and one where it equals the negative value)--once you have done that,
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all you need to do is solve each equation.
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So, I am going to solve, just using my usual techniques: subtract two from both sides, and that gives me 9x = 27.
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Over here, if I subtract 2 from both sides, I am going to get 9x = -31.
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Then, I am going to divide both sides by 9.
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OK, so you handle the absolute value equations the same way, even if it is more complex.
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The other thing to keep in mind is that sometimes, you have to first isolate the absolute value expression on the left side of the equation.
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For example, if you were given 3 times |5x + 4| equals 6, the first step would be to divide both sides by 3,
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because once you have done that, then you have the absolute value isolated,
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and you can proceed by saying 5x + 4 = 2 or 5x + 4 = -2, and then solving both of those.
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In some situations, some absolute value equations, where the absolute value equals c, if c is less than 0, these have no solution.
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So, if it says that the absolute value of x is a negative number, then there is no solution; and we just say that the solution is the empty set.
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And to make this more concrete: if I said that the absolute value of x equals -4...well, there is no situation
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where an absolute value is going to be a negative number, because remember: the absolute value is defined
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as the distance between that absolute value and 0 on the number line.
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And you can't have negative distance; so there is no solution here.
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And instead, we either just write the empty set as such, or like this, indicating that there is no solution.
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We just talked about a situation where there is no solution.
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An absolute value equation can have 0 solutions (which we just discussed--it is the empty set), 1 solution, or 2 solutions.
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And it is important to check each answer to make sure that it is a valid solution.
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For example, if I have |x + 4| = 9, I am going to go ahead and solve that,
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using the usual technique of turning it into two related equations, x + 4 = 9 and x + 4 = -9.
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I am going to solve that: x = 5; here, I am going to get x = -13.
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So, let's check this by going back to the original, |x + 4| = 9.
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If x = 5, then I am going to get |5 + 4| = 9; so, the absolute value of 9 equals 9--and that is true.
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Since this is true, this is a valid solution; x = 5 is a valid solution.
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OK, doing the same thing for my other solution: |x + 4| = 9...
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trying this |-13 + 4| = 9, well, -13 + 4 is -9, so the absolute value of -9 is 9.
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This is also a valid solution, since this is a true statement.
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So here, I have two solutions: previously, we discussed that, if you end up with something like |x| = -6, there is no solution; it is the empty set.
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The situation where you can get one solution is if you end up with |x| = 0.
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So, if I went to solve this, I would say, "x equals +0, and x equals -0"; but that doesn't really just make sense; they are just 0.
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Therefore, there is only one solution: x = 0; so here, I have one solution.
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Three possibilities: no solution, one solution, or two solutions.
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Or you may think you have two solutions, and then you go back and plug them in, and find out one is not valid.
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And in that case, you could have something like this, that appears that it would end up with two solutions,
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and it only ends up with one, or possibly even none.
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OK, Example 1: Evaluate for x = -3; and this is multiple absolute value terms here--
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substituting in -3 for x, solving for the absolute value of this would be 3 times -3...that is going to give me |-9 - 4|;
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here, I have 2 times -3; that is |-6 + 3|; minus 3 times...the absolute value of a negative, times a negative, is a positive.
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OK, this is |-9 - 4|; that is the absolute value of -13, plus |-6 + 3|; that is -3; minus 3...absolute value of 3.
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Now, I just need to find the absolute value for each of these.
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Well, the absolute value of -13 is 13; the absolute value of -3 is 3; the absolute value of 3 is 3;
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but this time, we are multiplying it times a -3; this is -3 times |3| (which is 3).
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So here, it's 13 + 3...-3 times 3 is -9, so 13 + 3 is 16, minus 9 is 7.
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OK, so we are solving this by substituting in -3 for x, finding the absolute values for these three, and then this one is multiplied by -3, and then simply adding.
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Example 2: I have an absolute value expression on the left, but it is not isolated.
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So, the first step is to isolate the absolute value, and then find the two related equations and solve them.
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Divide both sides by 4 to get that isolated.
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Now, recall that the absolute value of x equals 2, for example: this means that x could equal 2,
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or x could equal -2--to just illustrate the definition of absolute value.
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In order to get rid of these absolute value symbols, I am going to create two related equations,
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3x + 4 = 12, or 3x + 4 = -12: then I am going to solve both, and check to make sure that they are valid solutions.
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3x = 8; so divide both sides...I had 3x + 4, so I subtracted 4 from both sides.
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And then, over here, I have 3x = -16, subtracting 4 from both sides--I am just doing these together.
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Now, dividing both sides by 3 is going to give me x = 8/3; dividing here, I get -16/3.
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Now, check each of these in the original: that is this 4 times the absolute value of 3x + 4 equals 48.
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First checking this one: 4, and this is 3 times 8/3 + 4, equals 48...so the 3's cancel out, and that gives me 8 + 4...equals 48,
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so 4 times the absolute value of 12 equals 48; the absolute value of 12 is 12, so 4 times 12 equals 48; and it does, so this is valid.
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This first solution is valid.
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Now, substituting in -16/3, again, in this original: that is 4 times |3(-16/3) +4| = 48.
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The 3's cancel out; that gives me 4|16 + 4| = 48.
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-16 +4 is -12...equals 48...the absolute value of -12 is 12; so again, I come up with 4 times 12 equals 48, and that is valid.
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That is a true statement; so both of these solutions are valid.
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So this time, I had two solutions, 8/3 and -16/3, that satisfy this absolute value equation.
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Example 3: again, my first step is to isolate the absolute value expression on the left side of the equation.
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And I am going so start out by subtracting 15 from both sides, and that is going to give me -3 on the right.
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Now, I want to divide both sides by 3, and that is going to give me |2x - 4| = -1.
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And I don't even need to go any farther, because what this is saying is that
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the absolute value of whatever this expression ends up being (once I solve for x)--the absolute value of this equals -1.
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Well, that is not valid; you cannot have an absolute value equal a negative number, because it violates the definition of absolute value.
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Therefore, I don't even need to go any farther; I can just say that there is no solution, or that it is the empty set.
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So, there is no solution to this absolute value equation.
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The important thing is to just look carefully at your work and make sure you didn't make any mistakes.
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And if you did all the math correctly and handled this correctly, and you come up with something like this, then you didn't do anything wrong.
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It is just that there is no solution.
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OK, another absolute value expression: this time, the absolute value is already isolated on the left.
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So, looking at this, it looks more complex; but we just use the same logic that we did with the simple case.
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If the absolute value of x is 2, x equals 2, or x equals -2.
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So, I do the same thing here: I get rid of the absolute value bars, and this is my positive permutation.
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And then, I also have 2x - 7 = -(3x + 8).
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OK, solving each of these: I am going to add 7 to both sides; that is going to give me 2x = 3x +15.
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Then, I am going to subtract 3x from both sides, which is going to give me -x = 15.
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I am going to multiply both sides by -1 to get x = -15.
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So, that is my first solution: solving this...this is 2x - 7 = -3x - 8.
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Adding 7 to both sides is 2x = -3x -1.
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Adding 3x to both sides: 5x = -1; divide both sides by 5: x = -1/5.
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OK, check: with absolute value equations, you always have to check your solutions.
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So, checking this back in the original equation: 2 times -15, minus 7--the absolute value of that--equals 3 times -15, plus 8.
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OK, so I have 2(-15), which is -30, minus 7, equals 3(-15), which is -45, plus 8.
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This gives me |-37| equals...well, -45 + 8 is -37; the absolute value of -37 is 37.
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Well, 37 does not equal -37, so this is not a valid solution; this is not true--this did not satisfy this equation.
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So, x = -15 is not a valid solution; let's try this one, x = -1/5, substituting it in here.
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|2(-1/5) - 7| = 3(-1/5) + 8: that is going to give me |(-2/5)-7| = (-3/5) + 8.
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So then, adding these two together, I am going to get |-7 2/5| = 8 - 3/5...is 7 2/5.
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OK, the absolute value of -7 2/5 is 7 2/5, equals 7 2/5; and that is true; this is a valid solution.
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So, I actually have only one solution to this equation, and it is x = -1/5.
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We are starting out by breaking this into two related equations and removing the absolute bar,
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solving each, getting two solutions, and then checking and finding out that the first one is not valid, and that the second one is valid.
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That concludes this session for Educator.com, and I will see you for the next Algebra II lesson.