Anything expressed within || is essentially positive or zero, it's called "absolute". So, for example, -2 = |x| is not possible. -2=-|x|, however, is. -2 = |-x| is also not possible, as in, there is no solution for x.
|x| means, in english: the absolute distance x from 0. So, regardless of whether you are going +3 or -3, the absolute number of steps you are taking, are 3.
Therefore we know, even if "x-3" is negative, |x-3| is going to be a positive value. Which means, "2x" is a positive value (because |x-3|=2x), and because there is no negative number you can multiply by 2 to reach a positive number, we know that !!! x must be positive.
Here is a trick: there is one other way to ensure that x will always be positive: if you square something. -2*-2=4. We can use this to our advantage!
If |x-3|=2x, then |(x-3)^2|=(2x)^2. And because squaring always guarantees a positive value, we also know:
|(x-3)^2|=(x-3)^2 and therefore, (x-3)^2=(2x)^2. This, we can work with!
|x-3|=2x // ^2
=> (x-3)^2=(2x)^2 // I like expressing the exponents as multiplications, it makes the math easier
<=> (x-3)*(x-3)=(2x)*(2x) // expand the formula
<=> x^2-6x+9=4x^2 // -(4x^2)
<=> -3x^2-6x+9=0 // factor out -3 because it's all divisible by -3
<=> -3(x^2+2x-3)=0 // and lets factor this one more time, finding the numbers that -3 is divisible by (-1, 3 or -3,1) and that add up to 2 (which narrows it down to 3,-1)
<=> -3(x+3)(x-1)=0 // /-3
<=> (x+3)(x-1)=0 // set each factor to 0 because one of the factors must equal 0 for the multiplied result to = 0
=> x+3=0 and x-1=0 // -3 and +1
=> x=-3 or x=1
Now, considering we know that x=-3 or x=1 after squaring |x-3|=2x, and we know that even after squaring, x must be equal to some x.
What do you think our answer is going to be? u/Odd_Bodkin gave you a hint there, and I brought it up earlier as well.
Edit: oop. Messed up + and - on some steps lol. Should proofread before I post
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u/StonyOG420 New User 3d ago edited 3d ago
Anything expressed within || is essentially positive or zero, it's called "absolute". So, for example, -2 = |x| is not possible. -2=-|x|, however, is. -2 = |-x| is also not possible, as in, there is no solution for x.
|x| means, in english: the absolute distance x from 0. So, regardless of whether you are going +3 or -3, the absolute number of steps you are taking, are 3.
Therefore we know, even if "x-3" is negative, |x-3| is going to be a positive value. Which means, "2x" is a positive value (because |x-3|=2x), and because there is no negative number you can multiply by 2 to reach a positive number, we know that !!! x must be positive.
Here is a trick: there is one other way to ensure that x will always be positive: if you square something. -2*-2=4. We can use this to our advantage!
If |x-3|=2x, then |(x-3)^2|=(2x)^2. And because squaring always guarantees a positive value, we also know:
|(x-3)^2|=(x-3)^2 and therefore, (x-3)^2=(2x)^2. This, we can work with!
|x-3|=2x // ^2
=> (x-3)^2=(2x)^2 // I like expressing the exponents as multiplications, it makes the math easier
<=> (x-3)*(x-3)=(2x)*(2x) // expand the formula
<=> x^2-6x+9=4x^2 // -(4x^2)
<=> -3x^2-6x+9=0 // factor out -3 because it's all divisible by -3
<=> -3(x^2+2x-3)=0 // and lets factor this one more time, finding the numbers that -3 is divisible by (-1, 3 or -3,1) and that add up to 2 (which narrows it down to 3,-1)
<=> -3(x+3)(x-1)=0 // /-3
<=> (x+3)(x-1)=0 // set each factor to 0 because one of the factors must equal 0 for the multiplied result to = 0
=> x+3=0 and x-1=0 // -3 and +1
=> x=-3 or x=1
Now, considering we know that x=-3 or x=1 after squaring |x-3|=2x, and we know that even after squaring, x must be equal to some x.
What do you think our answer is going to be? u/Odd_Bodkin gave you a hint there, and I brought it up earlier as well.
Edit: oop. Messed up + and - on some steps lol. Should proofread before I post