Question

Solve the given inequality for real $x$ : $\dfrac{{(2x - 1)}}{3} \geqslant \dfrac{{(3x - 2)}}{4} - \dfrac{{(2 - x)}}{5}$

Answer 1

Hint: The addition is the sum of given two or more than two numbers, or variables and in addition, if we sum the two or more numbers then we obtain a new frame of the number will be found, also in subtraction which is the minus of given two or more than two numbers, but here comes with the condition that in subtraction the greater number sign represented in the number will stay constant example $2 - 3 = - 1$

Complete step by step answer:
Since given that the equation $\dfrac{{(2x - 1)}}{3} \geqslant \dfrac{{(3x - 2)}}{4} - \dfrac{{(2 - x)}}{5}$ 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 the usual cross multiplication of the given values, we have $\dfrac{{(2x - 1)}}{3} \geqslant \dfrac{{(3x - 2)}}{4} - \dfrac{{(2 - x)}}{5} \Rightarrow \dfrac{{(2x - 1)}}{3} \geqslant \dfrac{{5(3x - 2) - 4(2 - x)}}{{20}}$
By the multiplication operation, we have $\dfrac{{(2x - 1)}}{3} \geqslant \dfrac{{15x - 10 - 8 + 4x}}{{20}}$
Again, by the cross-multiplication, we get $\dfrac{{(2x - 1)}}{3} \geqslant \dfrac{{15x - 10 - 8 + 4x}}{{20}} \Rightarrow 20(2x - 1) \geqslant 3(15x - 10 - 8 + 4x)$
By subtraction we have $20(2x - 1) \geqslant 3(15x + 4x - 18)$ and also by the addition we have $20(2x - 1) \geqslant 3(19x - 18)$
Now by the multiplication, we get $40x - 20 \geqslant 57x - 54$
now Turing the variables on the left-hand side and also the numbers on the right-hand side we get $40x - 20 \geqslant 57x - 54 \Rightarrow 40x - 57x \geqslant - 54 + 20$ while changing the values on the equals to, the sign of the values or the numbers will change.
Again, by the addition and subtraction we get $ - 17x \geqslant - 34$, Now cancel the negative signs and the represented values get to change too
Then we get \[34 \geqslant 17x\]
Finally, by the division we get $2 \geqslant x$
The greater than or equals to inequality is the symbol represented as $ \geqslant $ which means the left-side entries or numbers are greater or equals values than the right-side entries like $x \geqslant y$ where either x is a greater value than y or x equals to y.
Thus we have the value as least possible is $ - \infty $ and highest possible is $2$ because which can be also expressed as $2 \geqslant x \Rightarrow x \leqslant 2$ hence we get the answer in the intervals as $x \in ( - \infty ,2]$

Note:
The other two operations are multiplication and division operations.
Since multiplicand refers to the number multiplied. Also, a multiplier refers to multiplying the first number. Have a look at an example; while multiplying $5 \times 7$the number $5$ is called the multiplicand and the number $7$ is called the multiplier. Like $2 \times 3 = 6$ or which can be also expressed in the form of $2 + 2 + 2(3times)$
The process of the inverse of the multiplication method is called division. Like $x \times y = z$ is multiplication thus the division sees as $x = \dfrac{z}{y}$. Like \[34 \geqslant 17x \Rightarrow \dfrac{{34}}{{17}} \geqslant x \Rightarrow 2 \geqslant x\]
Hence using simple operations, we solved the given problem.