## mathmath333 one year ago solve for $$x$$

1. mathmath333

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

2. 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}}

3. 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

4. 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}}

5. freckles

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

6. mathmath333

i havent studied intersection of sets

7. mathmath333

i just tried and posted

8. 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|

9. freckles

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

10. freckles

|dw:1436976374511:dw|

11. freckles

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

12. freckles

the reason x<-1 doesn't work is because x-1 is negative there the reason 1<x<2 doesn't work is because that isn't in the domain of the square root function

13. freckles

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

14. mathmath333

u mean the answer is $$x\geq 2$$

15. mathmath333

but wolfram also gives $$x=-1$$ which is not in ur intersecton

16. freckles

you need to include the zeros @mathmath333

17. freckles

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

18. 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

19. 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

20. freckles

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

21. freckles

22. freckles

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

23. freckles

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