Question

How do you solve $3x\le -\dfrac{5}{4}$ ?

Answer 1

Hint: We have been given an equation in one variable which includes an inequality to be dealt with. Thus, we have to find the interval in which this inequality holds true. First of all, we will separate all the constant terms on the right hand side and all the variable terms on the left hand side. Then we shall define the interval for which this inequality equation holds true.

Complete step by step solution:
Given that, $3x\le -\dfrac{5}{4}$
We shall first divide the left-hand side and the right-hand side of the equation by 3 to obtain our equation only in terms of variable-$x$.
$\Rightarrow \dfrac{3x}{3}\le -\dfrac{\dfrac{5}{4}}{3}$
$\Rightarrow x\le -\dfrac{5}{12}$
Now, we have to determine the interval on the number line for which this inequality holds true.
The symbol, $\le $ represents that out of the two quantities which are being compared, the one on the left-side is less than and equal to the quantity on the right-hand side.
This implies that $x$ is less than or equal to $-\dfrac{5}{12}$ and the hence $x$ lies in the interval $\left[ -\dfrac{5}{12},-\infty \right)$

Therefore, the solution of the equation $3x\le -\dfrac{5}{4}$ is $x\in \left[ -\dfrac{5}{12},-\infty \right)$.

Note: Generally, in mathematics, two kinds of brackets are used to write various interval values. The parentheses which are symbolized as (), are used to represent that the values enclosed within them are not included in that particular interval. However, the square brackets which are symbolized as [], are used to represent that the numerical values enclosed within them are included and have to be taken into consideration in that particular interval.