Solve the inequality -1 <= x^2 - 2x <= 8

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embizze eNotes educator| Certified Educator

Solve the inequality ` ``-1<=x^2-2x<=8 ` :

(a) `-1<=x^2-2x `

`0<=x^2-2x+1`

` ` `0<=(x-1)^2 `

This is true for all real values of x.

(b) `x^2-2x<=8 `

`x^2-2x-8<=0 `

`(x-4)(x+2)<=0 `

For x<-2 the left hand side is positive; try a test value like x=-3.

For -2<x<4 the left side is negative; try a test value like x=0.

For x>4 the left side is positive; try a test value like x=5.

So for `-2<=x<=4,(x-4)(x+2)<=0 `

Thus the correct solution is `-2<=x<=4 `

Here is a graph:

justaguide eNotes educator| Certified Educator

Excellent answer, embizze!