Question

Solve the inequality $-15<\dfrac{3\left( x-2 \right)}{5}\le 0$

Answer 1

Hint: In this problem we need to solve the given inequality or the expression that means we need to calculate the range of values for the variable $x$ for which the given inequality or the expression is satisfied. For this we will consider the term which is having the variable $x$ and write the different operations that are involved in that term. Now we will apply the reverse operations for the operation involved in the term one by one and simplify the expression to get the required result.

Complete step by step answer:
Given inequality or the expression is $-15<\dfrac{3\left( x-2 \right)}{5}\le 0$.
In the above expression the term $\dfrac{3\left( x-2 \right)}{5}$ has the variable $x$.
In the above term we can observe that the number $5$ is in division. So, we are going to multiply the whole expression with $5$, then we will get
$-15\times 5<\dfrac{3\left( x-2 \right)}{5}\times 5\le 0\times 5$
Simplifying the above expression by using basic mathematical operations, then we will have
$-75<3\left( x-2 \right)\le 0$
In the above expression we can observe that the number $3$ in multiplication. So, we are going to divide the whole expression with $3$, then we will get
$-\dfrac{75}{3}<\dfrac{3\left( x-2 \right)}{3}\le \dfrac{0}{3}$
Simplifying the above expression by using the basic mathematical operation. Then we will have
$-25 < x-2\le 0$
In the above expression we can observe that the number $2$ is in subtraction. So, we are going to add the number $2$ to whole expression, then we will get
$-25+2 < x-2+2\le 0+2$
Simplifying the above expression by using the basic mathematical operations, then we will have
$-23 < x\le 2$

Hence the solution of the given inequality $-15<\dfrac{3\left( x-2 \right)}{5}\le 0$ is $-23 < x\le 2$ or $x\in \left( -23, 2 \right] $.

Note: In this problem we have given the inequality and a monomial, so we will get the range of values for variable $x$ as solution. If we have given equality and a monomial then we will get only one value of $x$ as a solution.