Question

How do you solve rational equations \[\left[ {\dfrac{{x - 1}}{{x - 3}}} \right] = \left[ {\dfrac{2}{{x - 3}}} \right] \] ?

Answer 1

Hint: Here in this question, given a rational equation. We have solved the given expression to the simplest form. First, we can do the cross multiplication and later by using basic arithmetic operations. On simplification we have to solve the equation for the variable x to get the required solution.

Complete step by step solution:
A rational number is a number that can be expressed as a fraction where both the numerator and the denominator in the fraction are integers. The denominator in a rational number cannot be zero.
Expressed as an equation, a rational number is a number
 \[\dfrac{a}{b}\] , \[b \ne 0\]
where a and b are both integers.
This equation shows that all integers, finite decimals, and repeating decimals are rational numbers. In other words, most numbers are rational numbers.
Now consider the given expression
 \[ \Rightarrow \,\,\,\left[ {\dfrac{{x - 1}}{{x - 3}}} \right] = \left[ {\dfrac{2}{{x - 3}}} \right] \] -------(1)
Here, we have to solve this in the simplest form.
On applying the cross multiplication in equation (1), we get
 \[ \Rightarrow \,\,\,\left( {x - 1} \right)\left( {x - 3} \right) = 2\left( {x - 3} \right)\]
Remove all the parenthesis using multiplication, then
 \[ \Rightarrow \,\,\,x\left( {x - 3} \right) - 1\left( {x - 3} \right) = 2x - 6\]
 \[ \Rightarrow \,\,\,{x^2} - 3x - x + 3 = 2x - 6\]
On simplification, we get
 \[ \Rightarrow \,\,\,{x^2} - 4x + 3 = 2x - 6\]
Take all the term in RHS to LHS
 \[ \Rightarrow \,\,\,{x^2} - 4x + 3 - 2x + 6 = 0\]
 \[ \Rightarrow \,\,\,{x^2} - 6x + 9 = 0\]
Now, we have to solve the above equation for x by using the method of factorization.
Break the middle term as the summation of two numbers such that its product is equal to 9. Calculated above such two numbers are -3 and -3.
 \[ \Rightarrow \,\,\,{x^2} - 3x - 3x + 9 = 0\]
Making pairs of terms in the above expression
 \[ \Rightarrow \,\,\,\left( {{x^2} - 3x} \right) - \left( {3x - 9} \right) = 0\]
Take out greatest common divisor GCD from the both pairs, then
 \[ \Rightarrow \,\,\,x\left( {x - 3} \right) - 3\left( {x - 3} \right) = 0\]
Take \[\left( {x - 3} \right)\] common
Or
 \[ \Rightarrow \,\,\,{\left( {x - 3} \right)^2} = 0\]
Taking square root on both side, then
 \[ \Rightarrow \,\,\,x - 3 = 0\]
 \[ \Rightarrow \,\,\,x = 3\]
Hence, the required solution is x=3.
So, the correct answer is “ x=3.”.

Note: In mathematics rationalize means we have to solve the given rational number into the simplest form, rational numbers always in the form of \[\dfrac{p}{q}\] , but \[q \ne 0\] . Which can be solved by using any methods in mathematics but we get the required solution in the simplest form. That cannot be simplified further.