Question

How do you solve and graph the inequality $\left| 4-v \right| < 5$?

Answer 1

Hint: To solve the given inequality first we will solve the modulus given in the equation. As modulus of a number is its magnitude so when we solve the modulus of a number we will get the range of that number. By using this concept we will get the desired answer.

Complete step by step solution:
We have been given an inequality equation $\left| 4-v \right| < 5$.
We have to solve the given equation and draw a graph.
Now, let us first consider the modulus or LHS of the given equation. Then we will get
$\Rightarrow \left| 4-v \right|$
Now, we know that modulus of a number is its magnitude. Also we know that \[\left| a \right|=\pm a\]
Now, solving the above inequality we will get
$\begin{align}
  & \Rightarrow \left| 4-v \right| < 5 \\
 & \Rightarrow 4-v < \pm 5 \\
\end{align}$
Therefore the value of $4-v$ lies between $-5$ and 5
So we can write the above obtained equation as
$\Rightarrow -5 < 4-v < 5$
Now, simplifying the above obtained equation we will get
$\Rightarrow -5-4 < -v < 5-4$
Now, simplifying the above obtained equation we will get
$\Rightarrow -9 < -v < 1$
Now, multiplying the above obtained equation by negative sign we will get
$\begin{align}
  & \Rightarrow 9 > v > -1 \\
 & \Rightarrow -1 < v < 9 \\
\end{align}$
Hence we get the range of v as $-1$ to 9 by solving the given inequality.
Now, we need to graph the inequality. Then we will get

Note: The point to be remembered is that while solving the inequality doesn’t replace the inequality sign with an equal sign, it gives the incorrect solution. Also keep in mind that multiplying the inequality by negative sign changes the inequality sign.