Question

How do you solve $2 - 5\left| {5m - 5} \right| = - 73$?

Answer 1

Hint: In this question, we have to solve an expression that contains the absolute value. We can simplify the absolute value bar by splitting the expression into two cases. The first case is for the positive value, and the second case is for the negative value. Then simplify the expression to find the value of m.

Complete step-by-step answer:
In this question, we want to solve the expression,
$ \Rightarrow 2 - 5\left| {5m - 5} \right| = - 73$
Let us subtract 2 on both sides.
That is equal to,
$ \Rightarrow 2 - 2 - 5\left| {5m - 5} \right| = - 73 – 2$
Let us simplify the above expression.
$ \Rightarrow - 5\left| {5m - 5} \right| = - 75$
Now, multiply with -1 on both sides.
$\Rightarrow \left( { - 1} \right)\left( { - 5\left| {5m - 5} \right|} \right) = \left( { - 75} \right)\left( { - 1} \right)$
As we already know that multiplication of two negative numbers gives a positive answer.
Therefore,
$ \Rightarrow 5\left| {5m - 5} \right| = 75$
Let us divide both sides by 5.
That is equal to,
$ \Rightarrow \dfrac{{5\left| {5m - 5} \right|}}{5} = \dfrac{{75}}{5}$
So, we will get:
$ \Rightarrow \left| {5m - 5} \right| = 15$
Now, remove the absolute value term. This creates positive and negative signs on the right-hand sides of the expression. Because we know that $\left| x \right| = \pm x$ .
Therefore, when we will remove the absolute value term in the above expression:
$ \Rightarrow 5m - 5 = \pm 15$
First, we will take a positive sign on the right-hand side.
$ \Rightarrow 5m - 5 = 15$
Now, let us add 5 on both sides.
$ \Rightarrow 5m - 5 + 5 = 15 + 5$
That is equal to,
$ \Rightarrow 5m = 20$
Let us divide both sides by 5.
$ \Rightarrow \dfrac{{5m}}{5} = \dfrac{{20}}{5}$
Simplify the above expression.
$ \Rightarrow m = 4$
Now, we will take a negative sign on the right-hand side.
$ \Rightarrow 5m - 5 = - 15$
Now, let us add 5 on both sides.
$ \Rightarrow 5m - 5 + 5 = - 15 + 5$
That is equal to,
$ \Rightarrow 5m = - 10$
Let us divide both sides by 5.
$ \Rightarrow \dfrac{{5m}}{5} = \dfrac{{ - 10}}{5}$
Simplify the above expression.
$ \Rightarrow m = - 2$

Hence, the answer is 4 and -2.

Note:
An absolute value is also defined for the complex numbers, the quaternion, ordered rings, fields, and vector spaces. The absolute value is closely related to the notations of magnitude, distance, and norm in various mathematical and physical contexts.