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Suppose that 0 < | x-(-3) | < 2. Where does x have to be on...

Suppose that 0 < | x-(-3) | < 2. Where does x have to be on the x-axis?

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Answer #1

Given 0 < | x-(-3) | < 2

which is equivalent to 0 < | x+3 | < 2

Case (1) x+3 is positive i.e , x+3 > 0 . So, x > -3

Then 0 < x+3 < 2

Thus, 0-3 < x < 2-3

-3 < x < -1

Case (2) x+3 is negative i.e.,x < -3

Then 0 < -(x+3) < 2

0 < -x -3 < 2

0+3 < -x < 2+3

3 < -x < 5

Multiplying by -1 changed the inequality

-3 > x > -5

Thus, -5 < x < -3

From case (1) and (2) we have

x = (-3 , -1 ) U ( -5, -3) = (-5 , -3 ) U ( -3 , -1 )

So, x have to be between -5 and -1 excluding -3

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