Question

Solve the given inequality for real $x$: $\dfrac{x}{4} < \dfrac{{(5x - 2)}}{3} - \dfrac{{(7x - 3)}}{5}$

Answer 1

Hint: To solve linear inequality in one variable we will simplify it in similar way of linear equation and we will take variable terms to the one side and constants to the other side and then we will simplify it for value of variable(x) but we should keep in mind the notation of inequality which decides the interval of solution.

Complete answer:
Since given that the equation $\dfrac{x}{4} < \dfrac{{(5x - 2)}}{3} - \dfrac{{(7x - 3)}}{5}$ and then we need to find the value of the unknown variable $x$, so we will make use of the basic mathematical operations to simplify further.
Starting with the cross-multiplication method on the right-hand side, we have $\dfrac{x}{4} < \dfrac{{(5x - 2)}}{3} - \dfrac{{(7x - 3)}}{5} \Rightarrow \dfrac{x}{4} < \dfrac{{5(5x - 2) - 3((7x - 3))}}{{3 \times 5}}$
By the multiplication operation, we get $\dfrac{x}{4} < \dfrac{{25x - 10 - 21x - 9}}{{15}}$
By the use of the subtraction operation we have $\dfrac{x}{4} < \dfrac{{4x - 19}}{{15}}$
Again, by the cross-multiplication method, we get $\dfrac{x}{4} < \dfrac{{4x - 19}}{{15}} \Rightarrow 15x < 4(4x - 19)$
Again, by the multiplication we have $15x < 16x - 76$
Now Turing the variables on the left-hand side and also the numbers on the right-hand side we get $15x < 16x - 76 \Rightarrow 15x - 16x < - 76$ while changing the values on the equals to, the sign of the values or the numbers will change.
Hence, we have $ - x < - 76$
Now canceling the negative signs, we have $x > 76$ where $ - x < - y = x > y$
Thus, the unknown variable value is $x > 76$ which means the value of the variable can be at most of the number $76$

Note:
Generally students make mistakes while multiplying the inequality by minus 1 as they forget to change the sign of inequality. And sometimes students forget to check that boundary points are included or excluded.