Question

Identify the solution set for \[\left| 4-x \right|+1<3\].

Answer 1

Hint: In this problem, we have to find the solution set for the given inequality equation. Here we have the absolute value inequality, so we will have two cases, the positive case and the negative case by which we can split the bars of the absolute equation, we can then simplify the steps to find the value of x.

Complete step by step answer:
Here we have to find the solution set for the given inequality equation.
we know that the given inequality equation is,
\[\left| 4-x \right|+1<3\]
Here we have the absolute value inequality, so we will have two cases, the positive case and the negative case by which we can split the bars of the absolute equation.
\[\Rightarrow \left| 4-x \right|=4-x\], the positive case
\[\Rightarrow \left| 4-x \right|=-\left( 4-x \right)\], the negative case.
We can now solve both cases and find the value of x.
We can now take the positive case, we get
\[\Rightarrow 4-x+1<3\]
We can now subtract 5 on both sides, we get
\[\Rightarrow -x<-2\]
We can now multiply the minus sign on both sides, we get
\[\Rightarrow x>2\]
We can now take the negative case, we get
 \[\begin{align}
  & \Rightarrow -\left( 4-x \right)+1<3 \\
 & \Rightarrow -4+x+1<3 \\
\end{align}\]
We can now add 3 on both sides, we get
\[\Rightarrow x<6\]
Therefore, the solution set of \[\left| 4-x \right|+1<3\] is \[\left( 2,6 \right)\].

Note: We should remember that, if we have an absolute value equation, then we will have two cases, the positive case and the negative case, to remove the absolute bars and to solve for x. We should also know that, here we have to use the open brackets as we do not include the value of the numbers inside it.