Question

Solve 5x-3 < 7, when
(i) x is an integer
(ii) x is a real number

Answer 1

- Hint: Before solving this question, we must know the definitions of the two types of number systems that are as follows
INTEGERS:-
An integer is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75 and \[\sqrt{2}\] are not.
REAL NUMBERs:-
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The real numbers include all the rational numbers, such as the integer −5 and the fraction \[\dfrac{4}{3}\] , and all the irrational numbers, such as \[\sqrt{2}\] (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as \[\pi \] (3.14159265...).

Complete step-by-step solution -

Now, in this question, we will first solve the expression or the inequality and then we will find the value of x.
As mentioned in the question, we have to find the value of x when x belongs to different categories of numbers.
Firstly, we will solve the inequality as follows
\[\begin{align}
  & \Rightarrow 5x-3<7 \\
 & \Rightarrow 5x<7+3 \\
 & \Rightarrow 5x<10 \\
 & \Rightarrow x<\dfrac{10}{5} \\
 & \Rightarrow x<2 \\
\end{align}\]
Now, for part (i), we have x as an integer.
So, we can write as follows
\[x\in \left\{ 1,0,-1,-2...... \right\}\]
(Because integers are discrete entities)
Now, for part (ii), we have x as a real number.
So, we can write as follows
\[x\in \left( -\infty ,2 \right)\]
(Because there exists a real number between any two real numbers and hence, the set is continuous)
Hence, these are the values of x for the 2 parts as mentioned in the question.

Note:-It is important for the students to know what the difference is between integers and real numbers.
Also, it is important to know there exists a real number between any two real numbers and hence, the set is continuous in the second part.
Hence, these are the values of x for the 2 parts as mentioned in the question.
Also, the students can commit calculation errors, so, one should be extra careful while solving the question and while doing the calculations.