Question

Solve and express the solution in interval notation:
(x-3)^5 (x+2)^2 (1-x) < 0

Please explain your reasoning! :)

Answer 1

First check when the left side is equal to zero. This occurs when \(x=3\), \(x=-2\), and \(x=1\).

Plug in various points between these zero points. For example, \(x=0\) lies between 1 and -2, and the left side becomes \((-3)^5(2^2)(1)=-972\), which is negative. This means the interval \((-2,1)\) contains values of \(x\) that satisfy the inequality.

Answer 2

draw a number line with all of the zeros marked on it. (first step) this is because the expression is already in factored form

Answer 3

really, it is unnecessary to compute the values, only the sign

Answer 4

get the idea?

Answer 5

where the sign is negative, the function will be less than 0...
where the sign is positive, the function will be greater than 0.

Answer 6

So we test for one value between each zero to know if the numbers between the two zeros are positive or negative and find our solution that way? Huh
Thanks!

Answer 7

realy, you don't need to "test a value" you just observe whether each term will be positive or negative in that region, apply the appropriate exponent to each term and check the product. really, you onlu need to check if you have an even or odd power on the negatives but it's best to be thorough. but you need not compute the actual values, just see if the result is positive or negative.
does that make sense?

Answer 8

yeah don't worry i get it im not testing values just signs thank you!

Answer 9

you're welcome!!!

Answer 10

no worries here!