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How do you solve $-\dfrac{x}{x-3}\ge 0$ ?

seo-qna
Last updated date: 24th Jul 2024
Total views: 384.9k
Views today: 4.84k
Answer
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384.9k+ views
Hint: In this question, we will isolate the term $x$, on one side of the inequation by applying basic mathematical calculations to get the simplified value of x which is our final solution.

Complete step by step answer:
We have the given inequation as:
$-\dfrac{x}{x-3}\ge 0\to \left( 1 \right)$
$-\dfrac{x}{x-3}\ge 0$
Now since the left-hand side is in the form of a fraction, we will multiply both the sides with the term $(x-3)$to eliminate the fraction part.
On multiplying, we get:
$-\dfrac{x}{x-3}\times \left( x-3 \right)\ge 0\times \left( x-3 \right)$
Now on cancelling the terms in the right-hand side and simplifying the left-hand side, we get:
$-x\ge 0$
Now, multiply both the sides by -1, we get:
$x\le 0........$ (Since equality is reversed when we multiply negative values in an inequation)
Also note that x is undefined at 3.
$x\ne 3$
From the values of x above, we have these 3 intervals to test:
$\begin{align}
  & x\le 0 \\
 & 0\le x\le 3 \\
 & x\ge 3 \\
\end{align}$
Now, we have to test/check a point for each interval:
Therefore, for the interval $x\le 0$:
Let’s pick $x=-1$.
Therefore, on substituting on (1) we get:
$-\dfrac{-1}{-1-3}\ge 0$
On simplifying, we get:
$-0.25\ge 0$
Which is False. So drop this interval.
Now, for the interval $0\le x\le 3$:
Let’s pick $x=1$.
Therefore, on substituting on (1) we get:
$-\dfrac{1}{1-3}\ge 0$
On simplifying, we get:
$-0.5\ge 0$
Which is True. So keep this interval.

Now, for the interval $x\ge 3$:
Let’s pick $x=4$.
Therefore, on substituting on (1) we get:
$-\dfrac{4}{4-3}\ge 0$
On simplifying, we get:
$-4\ge 0$
Which is False. So drop this interval.
Therefore, from the above test, the required solution is:
$0\le x\le 3$

Note:
In the above question we have an inequation, which is different from the general what we call an equation. An inequation with a given condition may have a finite number of solutions. There is a common mistake which we tend to make while solving an inequation that is we convert inequation to the equation and then solve it and that is not a good practice because at the time we will multiply the equation by -1, equality won't be affected but inequality will reverse in an inequation. So we have to avoid this method to solve an inequation.