solve for \(x\)

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- mathmath333

solve for \(x\)

- chestercat

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- mathmath333

\(\large \color{black}{\begin{align} (x-1)\sqrt{x^2-x-2}\geq 0\hspace{.33em}\\~\\
\end{align}}\)

- mathmath333

i got
\(\large \color{black}{\begin{align} x\geq 1 \cup \left(x\geq 2 \cup x\leq -1\right)\hspace{.33em}\\~\\
\end{align}}\)

- freckles

\[x^2-x-2=(x-2)(x+1) \\ (x-1) \sqrt{(x-1)(x-2)} \ge 0\]
we should consider the domain of the square root function there
on that square root function's domain the square root function will be positive
so then you just need to consider when (x-1) is positive
and find the intersection of these sets

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## More answers

- mathmath333

how to solve this
\(\large \color{black}{\begin{align} x\geq 1 \cup \left(x\geq 2 \cup x\leq -1\right)\hspace{.33em}\\~\\
\end{align}}\)

- freckles

is that one symbol is suppose to be an intersect sign ?

- mathmath333

i havent studied intersection of sets

- mathmath333

i just tried and posted

- freckles

we should also go back to include when the function is 0
but I think you mean
\[(x \ge 1) \cap (x \ge 2 \cup x \le -1)\]
draw a number line
in keep in mind that x can be 1,-1,2 since that makes the function 0
but anyways it might helped to also graph that...
|dw:1436976303765:dw|

- freckles

notice the elements in common between x>=1
and also (x ge 2 union x<=-1)

- freckles

|dw:1436976374511:dw|

- freckles

|dw:1436976397456:dw|
but we also want to include these little endpoints
because at these endpoints we have the thingy is 0

- freckles

the reason x<-1 doesn't work is because x-1 is negative there
the reason 1

- freckles

anyways intersect just means you are trying to find the common elements

- mathmath333

u mean the answer is \(x\geq 2\)

- mathmath333

but wolfram also gives \(x=-1\) which is not in ur intersecton

- freckles

you need to include the zeros @mathmath333

- freckles

x>=2 or x=1 or x=-1

- freckles

this is what I meant by the statement "but we also want to include these little endpoints
because at these endpoints we have the thingy is 0
"|dw:1436977880095:dw|
and the drawing that went along with it

- freckles

if the thing was just
\[(x-1) \sqrt{(x+1)(x-2)} >0 \]
the answer would have been to just look at the interesections

- freckles

the intersection of when x>1 and the domain of the square root function

- freckles

but we also had a equal sign in there

- freckles

\[(x-1) \sqrt{(x+1)(x-2)} >0 \text{ gives the solution } x>2 \\ (x-1)\sqrt{(x+1)(x-2)}=0 \text{ when } x=\pm 1 \text{ or } x=2 \\ \]

- freckles

therefore the solution to
\[(x-1) \sqrt{(x+1)(x-2)} \ge 0 \text{ is } x>2 \text{ or } x= \pm 1 \text{ or } x=2 \\ \text{ clean it up a little } x \ge 2 \text{ or } x=\pm 1 \]

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