Question

How do solve the equation $\left| 3n-4 \right|=1$ ?

Answer 1

We can easily solve this type of equation by using the property of absolute values. We have to take two cases into consideration. In the first case we consider a positive value of the term $3n-4$ and in the second case we consider the term $3n-4$ to be a negative one and solve either of the equations to get the solutions of the given equation.

Complete step-by-step solution:
The expression we have is
$\left| 3n-4 \right|=1$
We solve this problem using the property of absolute values.
First, we remove the absolute value term in the given equation. This creates two cases among which the first one is where we consider the positive value of the term $3n-4$ and in the second one is where we consider the term $3n-4$ to be negative one.
Therefore, the given equation becomes
$3n-4=1......\left( \text{1} \right)$ , when $\left( 3n-4 \right)>0$
And $-\left( 3n-4 \right)=1......\left( 2 \right)$ , when $\left( 3n-4 \right)<0$
Considering equation $\left( 1 \right)$ we get
$\Rightarrow 3n-4=1$
Adding $4$ to both the sides of the above equation we get
$\Rightarrow 3n=1+4$
Further simplifying we get
$\Rightarrow 3n=5$
Dividing both sides of the above equation by $3$ we get
$\Rightarrow n=\dfrac{5}{3}$
Considering equation $\left( 2 \right)$ we get
$\Rightarrow -\left( 3n-4 \right)=1$
$\Rightarrow 3n-4=-1$
Adding $4$ to both the sides of the above equation we get
$\Rightarrow 3n=-1+4$
Further simplifying we get
$\Rightarrow 3n=3$
Dividing both sides of the above equation by $3$ we get
$\Rightarrow n=\dfrac{3}{3}$
$\Rightarrow n=1$
Also, we put both the solutions in the given equation. As both of the solutions satisfy the given equation, we can say that both the solutions are not extraneous.
Therefore, the solution of the given equation is $n=\dfrac{5}{3},\text{ }1$

Note: While removing the absolute term we have to take both the cases $\left( 3n-4 \right)>0$ and $\left( 3n-4 \right)<0$ into consideration to get all the solutions. Also, we must properly do the calculations while simplifying to avoid mistakes.