Question

Solve the inequality: $\dfrac{{3\left( {x - 2} \right)}}{5} \leqslant \dfrac{{5\left( {2 - x} \right)}}{3}$.

Answer 1

In the given question, we have to solve the inequality given to us in the problem itself. So, we have made use of the algebraic and simplification rules in order to simplify the inequality and find a solution to the inequality. One must know that we can multiply a positive number without changing the sign of inequality.

Complete step by step answer:
In the given question, we are given an inequality $\dfrac{{3\left( {x - 2} \right)}}{5} \leqslant \dfrac{{5\left( {2 - x} \right)}}{3}$.
Now, we know that $15$ is a positive number. So, multiplying both sides of the inequality by $15$ without changing the inequality signs, we get,
$\Rightarrow 15\left[ {\dfrac{{3\left( {x - 2} \right)}}{5}} \right] \leqslant 15\left[ {\dfrac{{5\left( {2 - x} \right)}}{3}} \right]$
Cancelling the common factors in the numerator and denominator, we get,
$\Rightarrow 9\left( {x - 2} \right) \leqslant 25\left( {2 - x} \right)$
Opening the brackets, we get,
$\Rightarrow 9x - 18 \leqslant 50 - 25x$

Now, we can simplify the inequality by isolating the variables and constants separately. So we shift the constant terms to the right side of the equation and all the variables to the left side of the equation. So, we get,
$\Rightarrow 9x + 25x \leqslant 50 + 18$
Adding up the like terms, we get,
$\Rightarrow 34x \leqslant 68$
Dividing both sides by $34$, we get,
$\Rightarrow x \leqslant \dfrac{{68}}{{34}}$
Cancelling the common factors in numerator and denominator, we get,
$\Rightarrow x \leqslant 2$
So, the solution set of the inequality given to us in the problem $\dfrac{{3\left( {x - 2} \right)}}{5} \leqslant \dfrac{{5\left( {2 - x} \right)}}{3}$ is the set of all the real numbers less than or equal to $2$.

Hence, we get the solution set as $\left( { - \inf ,2} \right]$.

Note: One must know that when multiplying both sides of the inequality by a positive integer does not change the sign of inequality but when we multiply both sides of the inequality by a negative number, the signs of inequality change. One should take care of the calculations and should recheck them to verify the final answer. Remember to reverse the sign of the terms while shifting from one side to another.