Question

How do you graph inequality\[\left| {4 - v} \right| < 5\]?

Answer 1

Hint: Inequality means to find a range, or ranges, of values that an unknown variable can take and still satisfy the in Inequality. Recall that the modules of a number are simply its magnitude, or absolute value. Regardless of its sign.

Complete step by step answer:
So we have an inequality equation to evaluate
\[\left| {4 - v} \right| < 5\]
As I had mentioned in the hint that modules of a number are simply its magnitude.
As I mentioned inequality is to find a range of values that can satisfy the equation.
As we know the mod functionality, When we open ant mod.
It will become;
\[\begin{aligned}
  \left| x \right| = a \\
   \Rightarrow x = \pm a \\
\end{aligned} \]
That means that the absolute value of \[4 - v\] is less than\[5\]. This means that \[4 - v\] must lie between\[5\;and\; - 5\].
We can write this as;
\[ \Rightarrow - 5 < 4 - v < 5\]
That’s what's called a double inequality. It is very simple to solve just treating it as two separate inequalities.
From the left side we will get;
\[ \Rightarrow \; - 5 < 4 - v\]
Now as we solve inequality, we just need to do that, first we need to subtract \[4\] both the sides.
\[\begin{aligned}
   \Rightarrow - 5 - 4 < 4 - v - 4 \\
   \Rightarrow \; - 9 < - v \\
   \Rightarrow \;9 > v \\
   \Rightarrow v < 9 \\
\end{aligned} \]
 [Negative sign will get cancel out both the side and inequality sign will change]
From the right side we will get;
\[ \Rightarrow 4 - v < 5\]
Now we will solve it in the same manner by subtracting \[4\] both sides.
\[\begin{aligned}
   \Rightarrow \;4 - v - 4 < 5 - 4 \\
   \Rightarrow - v < 1 \\
   \Rightarrow \;v > - 1 \\
\end{aligned} \]
 [Multiply (-) sign both the side and inequality sign will change]
We can write these both equations together as;
\[ \Rightarrow - 1 < v < 9\]
Now, we need to plot the graph of\[\left| {4 - v} \right| < 5\]. As we already solve the range for the question.
I.e. \[ - 1 < v < 9\]

Note: Whenever we solve the inequality type of question we always need to focus on the inequality sign especially when we deal with negative sign into such inequality. And when we multiply the equation with the negative sign the always inequality sign gets changed.