Question

How do we get graph of \[\dfrac{\left| w-1 \right|}{\left| w+\dfrac{3}{5} \right|}>1\]?

Answer 1

Hint: From the question we have been asked to find the graph of the given expression with some condition. For solving this we will take the expression given and simplify or solve it by removing the modulus and find the solutions in different cases and combine all the cases and get our solution. After getting the final condition then we will graph it in the coordinate plane and explain the graph. So, we proceed with our solution as follows.

Complete step by step answer:
From the given, we have,
\[\Rightarrow \dfrac{\left| w-1 \right|}{\left| w+\dfrac{3}{5} \right|}>1\]
Now, we will remove the modulus to the expression.
So, we get two cases when we remove the modulus. They are,
\[\Rightarrow \dfrac{w-1}{w+\dfrac{3}{5}}>1\] and \[\Rightarrow \dfrac{1-w}{w+\dfrac{3}{5}}>1\]
So, now we will solve these both cases and find the solution for them and see whether they are valid or not and proceed further.
So, now we will take the inequality,
\[\Rightarrow \dfrac{1-w}{w+\dfrac{3}{5}}>1\]
So, we will multiply with the denominator on both sides of the inequality.
\[\Rightarrow 1-w>w+\dfrac{3}{5}\]
\[\Rightarrow 1-\dfrac{3}{5}>2w\]
\[\Rightarrow \dfrac{1}{5}>w\]
Now, we will take the other case and solve it. So, we get the following.
\[\Rightarrow \dfrac{w-1}{w+\dfrac{3}{5}}>1\]
So, we will multiply with the denominator on both sides of the inequality.
\[\Rightarrow w-1>w+\dfrac{3}{5}\]
\[\Rightarrow -1>+\dfrac{3}{5}\]
This is a contradiction, it is not possible. So, this case will be eliminated from our answer.
For \[\Rightarrow \dfrac{1-w}{w+\dfrac{3}{5}}>1\] to happen, the denominator should be less than zero.
So, we get,
\[\Rightarrow w+\dfrac{3}{5}<0\]
\[\Rightarrow w<-\dfrac{3}{5}\]
So, the solution will be the union of \[\Rightarrow w<-\dfrac{3}{5}\] and \[\Rightarrow \dfrac{1}{5}>w\].
Therefore the solution is \[\Rightarrow -\dfrac{3}{5} < w < \dfrac{1}{5}\]
The graph for this will be as follows.

All points between point A and B on w-axis.

Note: Students should be very careful in doing the calculations. Students should know how to solve inequalities involving the modulus function. We must not forget that we should also take the step or consider the condition which is \[\Rightarrow w+\dfrac{3}{5}<0\] or our solution will be wrong.