Four pairs of inequalities are given below. For each pair of inequalities, decid

Question

Four pairs of inequalities are given below. For each pair of inequalities, decide whether inequality A implies inequality B, and whether inequality B implies inequality A, and briey justify your answers. If both inequalities imply each other, we will say that the two inequalities are equivalent

(a) Let x and y be real numbers.

Inequality A: x < y                     Inequality B: x5 < y5

(b) Let x and y be real numbers.

Inequality A: x < y                   Inequality B: 2x < 2y

(c) Let x and y be real numbers.

Inequality A: x < y                 Inequality B: x2 < y2

(d) Let x and y be real numbers.

Inequality A: x < y                 Inequality B: ex < ey
(e) Let x and y be positive real numbers.
Inequality A: x < y                                   Inequality B:1/ x<1/ y

Explanation / Answer

1) Yes inequality A implies B and B implies A i.e it is an euivalent inequality

As by Euclids axiom anything eual added or subtracted from both sides if the equation doesnt change the equation .

2) neither A implies B nor B implies A . As if anything negative is multiplied to both sides of the inequality changes the sign of inequality . For instance 2<3 but -2>-3

3) yes A implies B and B implies A i.e. its an equivalent inequality. As if we square any number or take its square root in an inequality . The inequality remain unchanged.

4) neither A implies B nor B implies A as if we reciprocate the numbers in inequality , the sign of inequality changes

For instance 2<3 but 1/2 = 0.5> 1/3=0.3

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