This is a sample question from my lesson. f(x) = √(20 - 2x) the lesson says the answer is x ≤ 10. In my textbook it says "the expression under a radical may not be negative." So, how do I change the sign and solve to get the answer above?

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This is a sample question from my lesson. f(x) = √(20 - 2x) the lesson says the answer is x ≤ 10. In my textbook it says "the expression under a radical may not be negative." So, how do I change the sign and solve to get the answer above?

Mathematics
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You know that you can NOT take the square root of a NEGATIVE. Correct?
yes, i know
Ok, so therefore the part inside the square root (the 20-2x), must be greater than or equal to 0. i.e: \(20-2x\ge0\)

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\(\color{blue}{1)}\) Add \(2x\) to both sides. \(\color{blue}{2)}\) Divide by \(2\) on both sides.
Oh I see! So the addition would take away the negative sign leaving us to solve for x as normal? and adding on the other side wouldn't change ≥ to ≤? like 20 - 2x ≥ 0 20 ≤ 2x 20/2 ≤ 2x/2 10 ≤ x?
incorrect
Follow me carefully. \(\large\color{black}{ \displaystyle 20-2x\ge0 \\[0.5em]}\) \(\large\color{black}{ \displaystyle 20-2x \color{blue}{+2x}\ge0\color{blue}{+2x} \\[0.5em]}\) The \(-2x\) and \(+2x\) will cancel each other out. \(\large\color{black}{ \displaystyle 20\ge2x \\[0.5em]}\) Divide both sides by 2 \(\large\color{black}{ \displaystyle 10\ge x \\[0.5em]}\)
So, the 10 is greater than (or equal to) x. ` (Not the other way around) `
oh i see! i get it now! thank you so much!
Ok, very nice.
You can logically check that x is NOT greater than 10, because if you choose for example x=11, then you get: 20-2(11) 20-22 -2 and thus, since the part inside the square root can NOT be negative,, you can tell that x is NOT greater than 10.
And you can logically see why x=10, or other x values that are smaller than 10 make the square root 0 or greater than 0 (which is perfectly valid for us).
i never knew to check it like that. That's very good to know! thanks :)

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