Question

Write five rational numbers greater than \[ - 2\].
A) \[ - \dfrac{3}{2},\,\dfrac{{ - 1}}{2}, - 1,\sqrt 5 ,\dfrac{1}{2}\]
B) \[ - \dfrac{3}{2},\,\dfrac{{ - 1}}{2}, - 1,0,\dfrac{1}{2}\]
C) \[ - \dfrac{3}{2},\,\dfrac{{ - 1}}{2}, - \sqrt 3 ,0,\dfrac{1}{2}\]
D) \[ - \dfrac{5}{2},\,\dfrac{{ - 1}}{2}, - 1,0,\dfrac{1}{2}\]

Answer 1

Hint:
Here in this question, we are asked to write five rational numbers greater than \[ - 2\]. Rational numbers are those numbers which can be expressed in the form of a fraction \[\dfrac{p}{q}\], where \[q \ne 0\]. To find the rational numbers we will analyze each number provided in the options of the question. Then we will compare each number with \[ - 2\], so that we can find which number is greater than \[ - 2\].

Complete step by step solution:
We have to find five rational numbers that are greater than \[ - 2\]. Now we know that there are infinite rational numbers between any two given numbers. Hence, there will be infinite numbers between \[ - 2\] and infinity. We have to find only five rational numbers. We are provided with five choices, out of which only one will be correct. Hence, we will analyze all the options, and the rational numbers provided to us in them, to find the correct answer. Let us begin by analyzing the choices for option A.
We will analyze each rational number step by step.
The first choice in the set is \[\dfrac{{ - 3}}{2}\].
We know that \[\dfrac{{ - 3}}{2} = - 1.5\]. This is a rational number. Also, \[ - 1.5 > - 2\]. Hence, this choice of option A is correct.
The second choice is \[\dfrac{{ - 1}}{2}\].
We know that \[\dfrac{{ - 1}}{2} = - 0.5\]. This is a rational number. Also, \[ - 0.5 > - 2\]. Hence, this choice of option A is correct.
The third choice is \[ - 1\].
This is a rational number. Also, \[ - 1 > - 2\]. Hence, this choice of option A is correct.
The fourth choice is \[\sqrt 2 \].
This is an irrational number. Hence, this choice of option A is not correct.
The fifth choice is \[\dfrac{1}{2}\].
We know that \[\dfrac{1}{2} = 0.5\]. This is a rational number. Also, \[0.5 > - 2\]. Hence, this choice of option A is correct.
Since we have one choice wrong hence, this is not the correct option.
We will now analyze the choices for option B.
The first choice in the set is \[\dfrac{{ - 3}}{2}\].
We know that \[\dfrac{{ - 3}}{2} = - 1.5\]. This is a rational number. Also, \[ - 1.5 > - 2\]. Hence, this choice of option B is correct.
The second choice is \[\dfrac{{ - 1}}{2}\].
We know that \[\dfrac{{ - 1}}{2} = - 0.5\]. This is a rational number. Also, \[ - 0.5 > - 2\]. Hence, this choice of option B is correct.
The third choice is \[ - 1\].
This is a rational number. Also, \[ - 1 > - 2\]. Hence, this choice of option B is correct.
The fourth choice is 0.
This is a rational number. Also, we know that \[0 > - 2\]. Hence, this choice of option B is correct.
The fifth choice is \[\dfrac{1}{2}\].
We know that \[\dfrac{1}{2} = 0.5\]. This is a rational number. Also, \[0.5 > - 2\]. Hence, this choice of option B is correct.
Since we have all choices correct hence, this is the correct option.
We will now analyze the choices for option C.
The first choice in the set is \[\dfrac{{ - 3}}{2}\].
We know that \[\dfrac{{ - 3}}{2} = - 1.5\]. This is a rational number. Also, \[ - 1.5 > - 2\]. Hence, this choice of option C is correct.
The second choice is \[\dfrac{{ - 1}}{2}\].
We know that \[\dfrac{{ - 1}}{2} = - 0.5\]. This is a rational number. Also, \[ - 0.5 > - 2\]. Hence, this choice of option C is correct.
The third choice is \[ - \sqrt 3 \].
This is an irrational number. Hence, this choice of option C is not correct.
The fourth choice is 0.
This is a rational number. Also, we know that \[0 > - 2\]. Hence, this choice of option C is correct.
The fifth choice is \[\dfrac{1}{2}\].
We know that \[\dfrac{1}{2} = 0.5\]. This is a rational number. Also, \[0.5 > - 2\]. Hence, this choice of option C is correct.
Since we have one choice wrong hence, this is not the correct option.
We will now analyze the choices for option D.
The first choice in the set is \[\dfrac{{ - 5}}{2}\].
We know that \[\dfrac{{ - 5}}{2} = - 2.5\]. This is a rational number. Also, \[ - 2.5 < - 2\]. This does not fulfill the required condition. Hence, this choice of option D is not correct.
The second choice is \[\dfrac{{ - 1}}{2}\].
We know that \[\dfrac{{ - 1}}{2} = - 0.5\]. This is a rational number. Also, \[ - 0.5 > - 2\]. Hence, this choice of option D is correct.
The third choice is \[ - 1\].
This is a rational number. Also, \[ - 1 > - 2\]. Hence, this choice of option D is correct.
The fourth choice is 0.
This is a rational number. Also, we know that \[0 > - 2\]. Hence, this choice of option D is correct.
The fifth choice is \[\dfrac{1}{2}\].
We know that \[\dfrac{1}{2} = 0.5\]. This is a rational number. Also, \[0.5 > - 2\]. Hence, this choice of option D is correct.
Since we have one choice wrong hence, this is not the correct option.
Thus, the correct option is B.

Note:
Rational numbers are those numbers which can be expressed in the form of a fraction \[\dfrac{p}{q}\], where \[q \ne 0\]. The difference between fractions and rational numbers, however, is that rational numbers are both positive and negative, unlike the fractions. Also, when the denominator is 1, rational numbers become integers.
In this solution, we can eliminate option A and C easily because they consist of \[\sqrt 5 \] and \[\sqrt 3 \] because they are irrational numbers. But as per the question we need to find rational numbers greater than \[ - 2\]. Therefore, we can skip checking option A and C which consist of some numbers that are irrational.