Question

How do you find the solution set for the equation $\left| x-6 \right|=7$?

Answer 1

Hint: To find the solution set of the above equation which is given as: $\left| x-6 \right|=7$. Now, the term $\left( x-6 \right)$ is written in the modulus and we know that when we remove modulus then the term which is equal to modulus will get positive or negative sign so to solve the given equation when we eliminate modulus sign then the equation will look as: $x-6=\pm 7$. Now, first take positive sign and then solve the equation and then take negative sign and solve the equation and hence, you will get the solutions.

Complete step by step solution:
The equation given in the above problem is as follows:
$\Rightarrow \left| x-6 \right|=7$
Now, we are going to remove the modulus sign and on eliminating the modulus sign then the number equal to modulus i.e. 7 will be written with two signs one positive and other negative. Hence, the above equation will look like:
$\Rightarrow x-6=\pm 7$
Now, we are taking positive sign first and after that we are going to solve the above equation as follows:
$\Rightarrow x-6=7$
Adding 6 on both the sides we get,
$\begin{align}
  & \Rightarrow x-6+6=7+6 \\
 & \Rightarrow x=13 \\
\end{align}$
Hence, we get one solution of the above modulus equation as $x=13$ and second solution we are going to find by taking negative sign as follows:
$\Rightarrow x-6=-7$
Adding 6 on both the sides of the above equation we get,
$\begin{align}
  & \Rightarrow x=-7+6 \\
 & \Rightarrow x=-1 \\
\end{align}$
Hence, we got the second solution of the above modulus equation as -1.

Hence, we have two solutions, one is 13 and other is -1.

Note: You can check whether the solutions you are getting are correct or not by satisfying those values of x in the given modulus equation.
Checking whether $x=13$ holds true by satisfying this value in the above equation we get,
$\begin{align}
  & \Rightarrow \left| x-6 \right|=7 \\
 & \Rightarrow \left| 13-6 \right|=7 \\
 & \Rightarrow \left| 7 \right|=7 \\
\end{align}$
And we know that if we apply modulus on any number either positive or negative then what we get outside is the positive number. Hence, L.H.S = R.H.S.
Checking whether $x=-1$ holds true by satisfying this value in the above equation we get,
$\begin{align}
  & \Rightarrow \left| -1-6 \right|=7 \\
 & \Rightarrow \left| -7 \right|=7 \\
 & \Rightarrow 7=7 \\
\end{align}$
As you can see that both the values of x hold true so the solutions we have found are correct.