jmc

Question

Consider the following six statements:

1.

x < x^{2} < x^{3}

2.

x < x^{3} < x^{2}

4.

x^{2} < x < x^{3}

8.

x^{2} < x^{3} < x

16.

x^{3} < x < x^{2}

32.

x^{3} < x^{2} < x

Enter the sum of the labels of statement that hold for some value of

x .

For example, if you think the statements with labels 2 and 8 hold for some value of

x,

then enter

2 + 8 = 10.

Accepted Answer

If x < -1, then x^3 < x < x^2.If x = -1, then x = x^3 = -1 and x^2 = 1. If -1 < x < 0, then x < x^3 < x^2.If x = 0, then x = x^2 = x^3 = 0. If 0 < x < 1, then x^3 < x^2 < x.If x = 1, then x = x^2 = x^3 = 1. If x > 1, then x < x^2 < x^3.Thus, the only statements that can hold are the ones with labels 16, 2, 32, and 1, and their sum is 51.