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Absolute Value Equations

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Today, we're going to talk about absolute value equations. So we'll start by reviewing the idea of absolute value, and then we'll do some examples solving absolute value equations and verifying their solutions on a graph. So let's start by reviewing the idea of absolute value.

Absolute value is defined as the distance a number is away from 0 on a number line, and we say that it's always non-negative. So for example, the absolute value of 3 would be the distance 3 is away from 0 on a number line, which would just be 3. And the absolute value of negative 3 would be the distance negative 3 is away from 0 on a number line, which is also 3. However, we also know that we're taking the absolute value of a negative number, we can think of it as just reversing the sign. So this becomes a negative, negative 3, which is just positive 3.

So we can define the absolute value of a number in two ways. So the absolute value of x is going to be equal to x if x is greater than or equal to 0. And the absolute value of x is going to be equal to negative x if x is less than 0.

So let's look at an example of an absolute value equation. I have the absolute value of x is equal to 4. So we'll solve this equation and verify the solution on its graph. So we have an example of a graph y is equal to the absolute value of x.

So for this equation, we have to consider both the positive and negative version of the expression that's in the absolute value. Because once we take the absolute value of something, both the positive and negative version of it could be equal to 4. So I have both that x is equal to 4 and that negative x is equal to 4. And once I write these two equations, then I don't need to include the absolute value brackets because I've accounted for both the positive and the negative expressions.

So now, looking at my two equations, I see that one solution is that x is equal to 4. And looking at my second equation, I can cancel out this negative by dividing by negative 1 on both sides, and I see that x is also equal to negative 4. We have our two solutions for our equation and we can verify that by looking for the-- on the graph, what are the values of x that would make y equal to 4?

So at y equals 4, if I go over to my graph, I see that that corresponds with x equals to negative 4. And y equals 4, on my graph, also corresponds to x is equal to positive 4. So my solutions are verified that they are correct.

So let's look at another example. I've got the absolute value equation the absolute value of 2x plus 3 equals 9, and the absolute value equation graphed y equals 2x plus 3. So I'm going to first solve this equation algebraically, and then verify the solution on the graph.

So if the absolute value of 2x plus 3 is equal to 9, then we know that 2x plus 3 can equal 9 or negative 2x plus 3 would equal 9. So we have one equation that uses the expression as is inside the absolute value bars, and we have one expression that is multiplied by negative 1. And so we can solve both of these equations to find our two solutions.

So to solve my first one, I'll start by subtracting 3 on both sides. This leaves me with 2x equals 6. Then I'll divide by 2. So I find my first solution-- x is equal to 3.

On the second side, I'm going to start by dividing by negative 1 to cancel out this negative sign in front. So this is 2x plus 3 equals negative 9. Then I'll subtract 3 from both sides as I did with the first equation, leaving me with 2x equals negative 12. Dividing by 2 on both sides, I have x equals negative 6 for my second solution.

And as I said, I can verify that on my graph. So I'm looking for the values of x that would make y equal to 9. So I see that when y is 9, I have x is equal to 3. And also, when y is 9, I have x is equal to negative 6. So I verified my two solutions.

So for my last example, I've got the equation absolute value of 5x minus 11 plus 4 equals 8 and the equation y equals the absolute value of 5x minus 11 plus 4. So I'm going to solve this equation algebraically and then verify the solutions on the graph. So here we have a term that's outside of our absolute value bars.

So we need to start by isolating our absolute value bars and the expression in between. So I'm going to cancel out of this plus 4 by subtracting 4 on both sides. So now I have the absolute value of 5x minus 11 equals 4.

So now I can write my two equations. 5x minus 11 equals 4. Or we could have the negative of 5x minus 11, and that would also equal 4. Solving both equations, I'm going to start by adding 11 to my first equation. That gives me 5x equals 15. And dividing by 5, I see that x is equal to 3 for my first solution.

For my second equation, I'm going to start by dividing by negative 1 on both sides. So this gives me 5x minus 11 equals negative 4. Then I'll add 11 to both sides, and I'm going to get that 5x is equal to 7. Dividing both sides by 5, I find that x is equal to approximately 1.4 for my second solution.

So on my graph, I can verify that. I'm looking for the values of x that make y equal to 8. So if I go to 8 on my y-axis, I see that y is 8 when x is approximately 1.4, and that y is also 8 when x is equal to 3.

So let's go over our key points from today. The absolute value of a number is its distance from 0 on the number line, and is always non-negative. When solving absolute value equations, we must consider both the positive and negative values of the expression inside the absolute value bars, because they can equal the same value once we take the absolute value. And when solving absolute value equations, the first step is to isolate the absolute value expression on one side of the equation.

So I hope that these key points and examples helped you understand a little bit more about absolute value equations. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.