Question

How do you solve the inequality $3x+2\le 1$ ?

Answer 1

Hint: For these kinds of questions, all we need is simple and basic mathematics. We just have to group all the variables of one kind onto one side and constants to the other side and do the necessary manipulations. When we have an equality, we can do the addition and subtraction like how we do when there is just an equal to sign. But division and multiplication should be done with care.

Complete step by step solution:
The given equality to us is $3x+2\le 1$. We have to find $x$.
So let us send $2$ onto the right hand side of the inequality from the left hand side of the inequality.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow 3x+2\le 1 \\
 & \Rightarrow 3x\le 1-2 \\
\end{align}$
Let us do the subtraction on the right hand side of the inequality.
Upon doing so, we get the following :
\[\begin{align}
  & \Rightarrow 3x+2\le 1 \\
 & \Rightarrow 3x\le 1-2 \\
 & \Rightarrow 3x\le -1 \\
\end{align}\]
Now let us divide the entire equation with $3$.
Upon doing so, we get the following :
\[\begin{align}
  & \Rightarrow 3x+2\le 1 \\
 & \Rightarrow 3x\le 1-2 \\
 & \Rightarrow 3x\le -1 \\
 & \Rightarrow x\le \dfrac{-1}{3} \\
\end{align}\]
When there is an inequality in the question that we have to solve , then we cannot really solve for a constant value of the variable in the question. We can only solve for the limits of the variable.
\[x\le \dfrac{-1}{3}\]indicates that the maximum value of $x$ is $\dfrac{-1}{3}$ and it can take up all the value left of $\dfrac{-1}{3}$.
So here , we basically found out the domain of $x$.

$\therefore $ Hence, the limit of $x$ that we found upon solving the inequality $3x+2\le 1$ is \[x\le \dfrac{-1}{3}\].

Note: Multiplication and division must be done with care when there inequality symbol as the inequality symbol changes whenever we are multiplying or dividing with a negative number. Just be careful while calculating as any mistake would change the limits of the variable in the question.