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

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embizze | High School Teacher | (Level 2) Educator Emeritus

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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:

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The solution of the inequality -1 <= x^2 - 2x <= 8 has to be determined.

The expression x^2 - 2x has to be greater than equal to -1 as well as less than or equal to 8.

-1 <= x^2 - 2x

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

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

=> x - 1 >= 0

=> x >= 1

x lies in the set `(1, oo) `

x^2 - 2x <= 8

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

=> x^2 - 4x + 2x - 8 <= 0

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

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

This is the case when either of the terms x + 2 and x - 4 is less than or equal to 0 and the other is greater than or equal to 0.

x + 2 < 0, x - 4 >= 0

=> x <= -2 and x >= 4 which is not possible

x + 2 >= 0 and x - 4 <= 0

=> x >= -2 and x <= 4

x lies in the set (-2, 4)

The intersection of `(1, oo)` and (-2, 4] is (1, 4).

The solution of the inequality `-1 <= x^2 - 2x <= 8` is (1, 4)

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Sorry for the errors in this response. Please refer the correct response posted by embizze.