Question

How do you solve the inequality $\left| x-3 \right|>7$?

Answer 1

Hint: To solve the given inequality in which modulus is given, we are going to first remove the modulus in the given inequality and modulus is eliminated by writing the given inequality as: $\left( x-3 \right)>7$ and $\left( x-3 \right)<-7$. Now, solve these two inequalities and find the value of x and take the union of these two solutions.

Complete step-by-step solution:
The inequality given in the above problem which we have to solve is as follows:
$\left| x-3 \right|>7$
Now, we are going to eliminate the modulus given in the above problem as follows:
$\left( x-3 \right)>7$
$\left( x-3 \right)<-7$
You might be thinking that when we eliminate modulus sign in the case of equality then equality remains same which we are shown below:
\[\begin{align}
  & \left| x-3 \right|=7 \\
 & \Rightarrow x-3=7; \\
 & \Rightarrow x-3=-7 \\
\end{align}\]
But we have changed the inequality sign in the above. This is the concept that you have to take into account that in the case of inequality when we remove modulus and put negative signs then inequality reverses.
Now, moving on to solve the inequalities which we have shown above as follows:
$\left( x-3 \right)>7$
$\left( x-3 \right)<-7$
Solving first inequality from the inequalities given above we get,
$\Rightarrow \left( x-3 \right)>7$
Adding 3 on both the sides of the above inequality we get,
$\begin{align}
  & \Rightarrow x-3+3>7+3 \\
 & \Rightarrow x>10 \\
\end{align}$
Now, writing the above solution in the interval form we get,
$\Rightarrow x\in \left( 10,\infty \right)$ ………….. (1)
Now, solving the second inequality we get,
$\Rightarrow \left( x-3 \right)<-7$
Adding 3 on both the sides of the above inequality we get,
$\begin{align}
  & \Rightarrow x-3+3<-7+3 \\
 & \Rightarrow x<-4 \\
\end{align}$
Writing the above solution in the interval form we get,
$\Rightarrow x\in \left( -\infty ,-4 \right)$ …….. (2)
Now, we are going to take the union of eq. (1) and (2) we get,
$x\in \left( -\infty ,-4 \right)\bigcup \left( 10,\infty \right)$
Hence, we have solved the given inequality as $x\in \left( -\infty ,-4 \right)\bigcup \left( 10,\infty \right)$.

Note: The point to be noted in the above solution is in writing the interval form of the above solutions in x. Let us consider the first solution which is equal to:
$\Rightarrow x>10$
Now, writing the above solution in the interval form we get,
$\Rightarrow x\in \left( 10,\infty \right)$
The point to be noted is that when there is no equality sign given in the inequation then we have to put an open bracket which we have shown above. Now, let us consider the case when equality sign is also present along with the inequality:
$\Rightarrow x\ge 10$
Now, the solution of the above in the interval form is given as:
$x\in [10,\infty )$
As you can see we have changed the bracket from closed to an open bracket so make sure you will keep this subtle point taken into account.