Absolute value equations
Absolute Value Equation Example 2 Example of solving an absolute value equation
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- So we're given the following equation, that 5 times the
- absolute value of x plus 3 minus 3 is equal to 7.
- It's always a little daunting to see an equation with an
- absolute value sign.
- And when you only have one of them, like this, what I like
- to do is isolate it, and just kind of think it through from
- that point.
- So let's try to isolate the absolute
- value of x plus 3 here.
- Let's try to isolate that part of our equation.
- So the first thing we might want to do is add 3 to both
- sides of the equation.
- That'll get rid of this minus 3, or this negative 3, on the
- left-hand side.
- So let's add 3 to both sides of this equation.
- And this'll turn the equation into-- do it in that same pink
- color --the left-hand side will still be 5 times the
- absolute value of x plus 3.
- The minus 3, or the negative 3 plus the 3,
- those will cancel out.
- That'll just be 0.
- And then that will be equal to 7 plus 3, which is 10.
- Now, we have 5 times the thing we want to isolate.
- The best way to isolate it completely is to divide both
- sides of this equation by 5.
- So if you divide that side by 5, and then the
- right-hand side by 5.
- We divide it by 5 so that these guys will cancel out.
- 5 times something divided by 5 is just that something.
- So these cancel out.
- That's 2.
- So we're left with the absolute value of x plus 3 is
- equal to 10 divided by 5, which is 2.
- So we've simplified the equation a good bit, now we
- just have to put our thinking caps on a little bit.
- If the absolute value of something is 2, what does it
- mean that that something is?
- What are the two numbers that, if I were take it's absolute
- value, I could get 2.
- I'll do a little thing on the side here.
- We know that the absolute value of 2 is equal to 2.
- So maybe this thing was equal to 2.
- So maybe x plus 3 is equal to 2.
- If x plus 3 is equal to 2, and you take its absolute value,
- you're going to get 2 again.
- But we also know that the absolute value of negative 2
- is also equal to 2.
- So maybe x plus 3 is equal to negative 2.
- Because if x plus 3 is equal to negative 2, and we take its
- absolute value, then we're going to get 2 again.
- So we could also write or x plus 3 could be equal to
- negative 2.
- That's what I mean about thinking
- about it a little bit.
- Another way to think about it-- I've said this in other
- videos --is absolute value means distance from 0.
- So if we were to draw a number line here, that is 0, this is
- saying that, whatever this quantity is inside the
- absolute value sign, its distance from 0 is 2.
- So what numbers are 2 away from 0?
- Well, you have positive 2.
- I'll write a positive there explicitly.
- And you also have a negative 2.
- So this thing here could be a positive 2, or it could be a
- negative 2.
- Either way you take the absolute value of a positive 2
- or a negative 2, you're going to get a 2.
- So let's solve these.
- So over here we can subtract 3 from both
- sides of this equation.
- So if you subtract 3 from both sides, you get-- The left-hand
- side, you're just left with an x.
- These 3's cancel out.
- That's the whole point of subtracting the 3.
- x will be equal to 2 minus 3 is negative 1.
- So that is one solution to our absolute value equation.
- And what's our other solution?
- Well here, once again, let's subtract 3 from both sides.
- So you subtract 3 from both sides.
- The left-hand side just becomes an x.
- The right-hand side, negative 2 minus 3 is negative 5.
- So this is our other solution.
- And let's verify that they both work.
- So if x is equal to negative 1, what does
- this equation become?
- We have 5 times the absolute value of negative 1
- plus 3 minus 3.
- And if this really is a solution, then this should be
- equal to positive 7.
- So let's see.
- This is 5 times negative 1 plus 3 is 2.
- So it's 5 times the absolute value of 2 minus 3.
- The absolute value of 2 is just 2.
- So this is 5 times 2 minus 3.
- 5 times 2 is 10 minus 3, which is, indeed, equal to 7.
- So this is definitely a solution.
- Let's try the other one out. x is equal to negative 5.
- Running out of some real estate, but I could clear some
- up right here.
- If x is equal to negative 5, same drill.
- 5 times the absolute value of negative 5 plus 3 minus 3.
- This again should be equal to 7.
- Negative 5 plus 3 is negative 2.
- Since there's 5 times the absolute value of negative 2
- minus 3, the absolute value of negative 2 is positive 2.
- So this is 5 times 2 minus 3, which is 10 minus 3, which is
- again equal to 7.
- And notice, in this situation, the thing in the absolute
- value sign became a negative 2, because we said, oh, x plus
- 3 could be equal to negative 2.
- In the magenta situation, the thing in the absolute value
- sign was positive 2, because this was a situation where we
- assumed that the thing in the absolute value sign was a
- positive 2.
- Now if we were to graph these solutions on the number line,
- if we were to graph the two solutions on a number line--
- I'll draw a new number line here.
- Let me draw a new number line, just like that.
- Both of these are less than 0, so I'll put 0 all
- the way over here.
- Maybe 0, 1, 2.
- Let's go negative, negative 1, negative 2, negative 3,
- negative 4, negative 5, negative 6.
- This solution right here-- x is equal to negative 1 --we
- could plot on the number line right over there.
- And then the other solution-- x is equal to negative 5 --we
- can plot it right over there.
- Hopefully you found that enjoyable.
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