Question

How do you solve $\left| 3n-2 \right|=\dfrac{1}{2}$ ?

Answer 1

Hint: This problem contains the absolute value function. The absolute value function transforms any negative or positive term into its positive form. Therefore, we must solve the term within the absolute value for both its negative and positive equivalent.

Complete step by step answer:
Now considering from the question we have $\left| 3n-2 \right|=\dfrac{1}{2}$ .
Here in this question we have the absolute modulus so we have to solve for both positive and negative versions of the equation.
First we solve the given equation by equating it to the negative equivalent,
By equating it to the negative equivalent we get, $3n-2=-\dfrac{1}{2}$
We can solve this by using some simple transformations.
First, add $2$ on both sides of the above equation. By adding $2$ on both sides of the equation we get, $3n-2+2=-\dfrac{1}{2}+2$
By simplifying the above equation we get,
$3n=\dfrac{-1+4}{2}$
$\Rightarrow 3n=\dfrac{3}{2}$
$\Rightarrow n=\dfrac{1}{2}$
Therefore by equating the given equation to the negative equivalent we get the value of $n=\dfrac{1}{2}$
Now, we have to equate the given equation to the positive equivalent.
By equating the given equation to the positive equivalent we get the below equation, $3n-2=\dfrac{1}{2}$
We can solve the above equation by using some certain transformations.
To solve the above equation, first add $2$ on both sides of the above equation.
By adding $2$ on both sides of the equation we get, $3n-2+2=\dfrac{1}{2}+2$
By simplifying the above equation we get,
$3n=\dfrac{1+4}{2}$
$\Rightarrow 3n=\dfrac{5}{2}$
$\Rightarrow n=\dfrac{5}{6}$
By equating the given equation to the positive equivalent, we get the value of $n=\dfrac{5}{6}$
Therefore, the solutions are $n=\dfrac{1}{2}\text{ and }n=\dfrac{5}{6}$

Note:
We should be well aware of the absolute value function and its usage. We should have to do more practice on these types of problems. So we can solve the problems easily. We should be very careful while doing the calculation especially while equating to the negative equivalent and positive equivalent. If we have made some kind of mistake and done only one version of the expression then we will have only one solution which is a wrong conclusion.