Question

How do you solve $\dfrac{{x - 4}}{{x + 5}} < 0$?

Answer 1

Hint: In this question we will isolate the term $x$, so we will multiply the denominator term on both sides and doing some simplification we get the final solution.

Complete step-by-step solution:
We have the given inequation as:
$ \Rightarrow \dfrac{{x - 4}}{{x + 5}} < 0$
Now since the left-hand side is in the form of a fraction, we will multiply both the sides with the term $(x + 5)$ to eliminate the fraction part.
On multiplying, we get:
$ \Rightarrow \dfrac{{x - 4}}{{x + 5}} \times (x + 5) < 0 \times (x + 5)$
Now on cancelling the terms in the right-hand side and simplifying the left-hand side, we get:
$ \Rightarrow x - 4 < 0$
On transferring the term $4$ across the lesser than sign, we get:
$ \Rightarrow x < 4$, which is the required solution.

x < 4 is the required answer.

Note: In the question we have an inequation, which is different from the general what we call an equation.
An equation equates both the terms on the left-hand side and the right-hand side equally, whereas in inequalities, the left-hand side and right-hand side are not the same to each other.
Inequations can have the greater than sign which is $ > $ and the lesser than sign which is $ < $.
There can also be the greater than or equal to sign, which is $ \geqslant $ and the lesser than or equal to sign which is $ \leqslant $.
The solution $x < 4$ implies that the value of $x$ should be less than $4$, it implies that as soon as the value of $x$ is written as more than $4$, the term on the left-hand side won’t be lesser than $0$.