Question

How do you solve $\left| -10x-5 \right|=0$?

Answer 1

Hint: The given equation has a modulus sign on the LHS. We first need to remove this modulus sign according to the definition of the modulus function. The modulus function, when applied on a number, returns its absolute value, that is, it returns a non-negative number. Therefore, if $-10x-5\ge 0$, then $\left| -10x-5 \right|=-10x-5$. But if $-10x-5>0$ then $\left| -10x-5 \right|=-\left( -10x-5 \right)$. So our solution will be divided into two parts for the two intervals of $x$ on solving which we will get the required solutions.

Complete step by step solution:
The given equation is
$\Rightarrow \left| -10x-5 \right|=0$
We know that the modulus function returns the absolute value of the argument passed to it. So according to the argument $-10x-5$, we divide the solution into two cases.
Case I: $-10x-5\ge 0$
$\Rightarrow -10x-5\ge 0$
Adding $5$ both the sides
$\begin{align}
  & \Rightarrow -10x-5+5\ge 0+5 \\
 & \Rightarrow -10x\ge 5 \\
\end{align}$
Dividing both sides by $10$
$\begin{align}
  & \Rightarrow -x\ge \dfrac{5}{10} \\
 & \Rightarrow -x\ge \dfrac{1}{2} \\
\end{align}$
Multiplying by $-1$ and reversing the inequality sign
$\Rightarrow x\le -\dfrac{1}{2}$
Since the argument to the modulus is positive, so for this case we will have $\left| -10x-5 \right|=-10x-5$. Therefore, the given equation can be written as
$\Rightarrow -10x-5=0$
Adding $5$ both the sides, we get
$\begin{align}
  & \Rightarrow -10x-5+5=0+5 \\
 & \Rightarrow -10x=5 \\
\end{align}$
Dividing both the sides by $-10$ we get
$\begin{align}
  & \Rightarrow \dfrac{10x}{-10}=\dfrac{5}{-10} \\
 & \Rightarrow x=-\dfrac{1}{2} \\
\end{align}$
So the solution in this case is $x=-\dfrac{1}{2}$.
Case II: $-10x-5<0$
$\Rightarrow -10x-5\ge 0$
Adding $5$ both the sides
$\begin{align}
  & \Rightarrow -10x-5+5<0+5 \\
 & \Rightarrow -10x<5 \\
\end{align}$
Diving both sides by $10$
\[\begin{align}
  & \Rightarrow -x<\dfrac{5}{10} \\
 & \Rightarrow -x<\dfrac{1}{2} \\
\end{align}\]
Multiplying by $-1$ and reversing the inequality sign
$\Rightarrow x>-\dfrac{1}{2}$
In this case the argument to the modulus function is negative, so we will have
$\begin{align}
  & \Rightarrow \left| -10x-5 \right|=-\left( -10x-5 \right) \\
 & \Rightarrow \left| -10x-5 \right|=10x+5 \\
\end{align}$
Therefore, the given equation for this case will be written as
$\Rightarrow 10x+5=0$
Subtracting $5$ from both the sides, we get
$\begin{align}
  & \Rightarrow 10x+5-5=0-5 \\
 & \Rightarrow 10x=-5 \\
\end{align}$
Dividing both sides by $10$ we get
$\begin{align}
  & \Rightarrow \dfrac{10x}{10}=-\dfrac{5}{10} \\
 & \Rightarrow x=-\dfrac{1}{2} \\
\end{align}$
But this value is not matching with the condition $x>-\dfrac{1}{2}$. So there is no solution for $x>-\dfrac{1}{2}$.
Hence, the final solution for the given equation is $x=-\dfrac{1}{2}$.

Note:
Do not forget to reverse the inequality sign while multiplying an inequality by $-1$. We can also solve the given equation without dividing it into cases. This is because the RHS of the given equation is zero, and so the given equation $\left| -10x-5 \right|=0$ will get reduced to $-10x-5=\pm 0$, or equivalently, $-10x-5=0$. From here we will directly obtain $x=-\dfrac{1}{2}$.