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Question: How do you solve \(-\dfrac{x}{x-3}\ge 0\) ?...

How do you solve −xx−3≥0-\dfrac{x}{x-3}\ge 0 ?

Explanation

Solution

In this question, we will isolate the term xx, 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:
−xx−3≥0→(1)-\dfrac{x}{x-3}\ge 0\to \left( 1 \right)
−xx−3≥0-\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)(x-3)to eliminate the fraction part.
On multiplying, we get:
−xx−3×(x−3)≥0×(x−3)-\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≥0-x\ge 0
Now, multiply both the sides by -1, we get:
x≤0........x\le 0........ (Since equality is reversed when we multiply negative values in an inequation)
Also note that x is undefined at 3.
x≠3x\ne 3
From the values of x above, we have these 3 intervals to test:
x≤0 0≤x≤3 x≥3 \begin{aligned} & x\le 0 \\\ & 0\le x\le 3 \\\ & x\ge 3 \\\ \end{aligned}
Now, we have to test/check a point for each interval:
Therefore, for the interval x≤0x\le 0:
Let’s pick x=−1x=-1.
Therefore, on substituting on (1) we get:
−−1−1−3≥0-\dfrac{-1}{-1-3}\ge 0
On simplifying, we get:
−0.25≥0-0.25\ge 0
Which is False. So drop this interval.
Now, for the interval 0≤x≤30\le x\le 3:
Let’s pick x=1x=1.
Therefore, on substituting on (1) we get:
−11−3≥0-\dfrac{1}{1-3}\ge 0
On simplifying, we get:
−0.5≥0-0.5\ge 0
Which is True. So keep this interval.

Now, for the interval x≥3x\ge 3:
Let’s pick x=4x=4.
Therefore, on substituting on (1) we get:
−44−3≥0-\dfrac{4}{4-3}\ge 0
On simplifying, we get:
−4≥0-4\ge 0
Which is False. So drop this interval.
Therefore, from the above test, the required solution is:
0≤x≤30\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.