Question

How do you solve $2\left( 3x-5 \right)<4$?

Answer 1

Hint: We first try to find the similarities for binary operations in between equation and inequation. We try to keep the variable and the constants separate and on opposite sides of the inequality. We first complete the multiplication. Then we add 10 on both sides of the inequation and find the range of the variable $x$.

Complete step by step solution:
We have been given an inequality where the inequation is $2\left( 3x-5 \right)<4$.
We need to find the solution or range for the variable $x$.
We complete the multiplication of $2\left( 3x-5 \right)$ to get $2\left( 3x-5 \right)=6x-10$.
Now we try to keep the variable and the constants separate. We add 10 on both sides of the inequation
The binary operation of subtraction in case of inequation works in a similar way to an equation.
There will be no change in the sign of the inequation.
Now we add 10 to the both sides of the inequation of $6x-10<4$.
\[\begin{align}
  & 6x-10<4 \\
 & \Rightarrow 6x-10+10<4+10 \\
\end{align}\]
We perform the binary operations in both cases and get
\[\begin{align}
  & 6x-10+10<4+10 \\
 & \Rightarrow 6x<14 \\
\end{align}\]
We now divide with 6 to get
\[\begin{align}
  & \dfrac{6x}{6}<\dfrac{14}{6} \\
 & \Rightarrow x<\dfrac{7}{3} \\
\end{align}\]
The solution for the equation $2\left( 3x-5 \right)<4$ is that the value of the variable $b$ will be less than \[\dfrac{7}{3}\].

Note: We need to remember that in inequality we are never going to find a particular solution. In most of the cases we are going to find the range or interval as a solution. For our given inequality $2\left( 3x-5 \right)<4$, we got a variable $x$ to be less than \[\dfrac{7}{3}\]. It’s a range of $x$. We can express it as $x\in \left( -\infty ,\dfrac{7}{3} \right)$.