Question

01. Let π = {β2; 1; 0;1/2;3/4; 2;10/3; 4; β5}. Explain the elements of each of the following sets: a) {π₯ β π/π₯ β 4 < 0} b) {π₯ β π/π₯ 2 β 6π₯ + 8 < 0} c) {π₯ β π/4 β π₯ β€ 0} d) {π₯ β π/π₯ 2 + 1 β€ 0}

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Given set π = {β2; 1; 0; 1/2; 3/4; 2; 10/3; 4; β5}

a) {π₯ β π/π₯ - 4 < 0}

To solve this, first, we need to find the elements in π where π₯ - 4 < 0.

π₯ - 4 < 0

π₯ < 4

The elements in set π that satisfy π₯ < 4 are {-2; 1; 0; 1/2; 3/4; 2, 10/3}.

b) {π₯ β π/π₯^2 - 6π₯ + 8 < 0}

To solve this, first, we need to find the elements in π where π₯^2 - 6π₯ + 8 < 0.

(π₯ - 4)(π₯ - 2) < 0

The solutions to this inequality are π₯ β (2, 4).

Thus, the elements in set π that satisfy this inequality are {3/4}.

c) {π₯ β π/4 - π₯ β€ 0}

To solve this, first, we need to find the elements in π where 4 - π₯ β€ 0.

4 - π₯ β€ 0

π₯ β₯ 4

The only element in set π that satisfies this inequality is {4}.

d) {π₯ β π/π₯^2 + 1 β€ 0}

To solve this, first, we need to find the elements in π where π₯^2 + 1 β€ 0.

This inequality has no real solutions because π₯^2 + 1 is always greater than 0 for all real values of π₯.

Therefore, there are no elements in π that satisfy this inequality.

\textbf{Answer:}

a) {-2; 1; 0; 1/2; 3/4; 2; 10/3}

b) {3/4}

c) {4}

d) No elements in setπ satisfy the inequality.

a) {π₯ β π/π₯ - 4 < 0}

To solve this, first, we need to find the elements in π where π₯ - 4 < 0.

π₯ - 4 < 0

π₯ < 4

The elements in set π that satisfy π₯ < 4 are {-2; 1; 0; 1/2; 3/4; 2, 10/3}.

b) {π₯ β π/π₯^2 - 6π₯ + 8 < 0}

To solve this, first, we need to find the elements in π where π₯^2 - 6π₯ + 8 < 0.

(π₯ - 4)(π₯ - 2) < 0

The solutions to this inequality are π₯ β (2, 4).

Thus, the elements in set π that satisfy this inequality are {3/4}.

c) {π₯ β π/4 - π₯ β€ 0}

To solve this, first, we need to find the elements in π where 4 - π₯ β€ 0.

4 - π₯ β€ 0

π₯ β₯ 4

The only element in set π that satisfies this inequality is {4}.

d) {π₯ β π/π₯^2 + 1 β€ 0}

To solve this, first, we need to find the elements in π where π₯^2 + 1 β€ 0.

This inequality has no real solutions because π₯^2 + 1 is always greater than 0 for all real values of π₯.

Therefore, there are no elements in π that satisfy this inequality.

a) {-2; 1; 0; 1/2; 3/4; 2; 10/3}

b) {3/4}

c) {4}

d) No elements in set

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