How do you solve \[\dfrac{1}{2}\left( 4x+6 \right)=\dfrac{1}{3}\left( 9x-24 \right)\]?

Hint: From the above given question, we have been asked to solve \[\dfrac{1}{2}\left( 4x+6 \right)=\dfrac{1}{3}\left( 9x-24 \right)\] .We can solve the above-given equation from the question given by using some simple transformations and substitutions. By using some transformations, we can get the equation more simplified, and then we can do the calculation very easily.

Complete step by step answer:
From the question, it has been already given that \[\dfrac{1}{2}\left( 4x+6 \right)=\dfrac{1}{3}\left( 9x-24 \right)\]
Now, as we have already discussed above, we have to use some simple transformations to get the equation more simplified.
Here, what we have to do is, multiply the terms within the parenthesis.
By multiplying the terms within the parenthesis,
We get the equation as below
\[\dfrac{1}{2}\left( 4x+6 \right)=\dfrac{1}{3}\left( 9x-24 \right)\]
\[\Rightarrow \left( \dfrac{1}{2}.4x \right)+\left( \dfrac{1}{2}.6 \right)=\left( \dfrac{1}{3}.9x \right)-\left( \dfrac{1}{3}.24 \right)\]
On furthermore simplifying the above equation, we get \[2x+3=3x-8\]
Now, solve for \[x\]while keeping the equation balanced.
Solving for \[x\] while keeping the equation balanced, we get
\[2x+3-2x+8=3x-8-2x+8\]
\[\Rightarrow 3+8=3x-2x\]
\[\Rightarrow x=11\]
So, as we have been already discussed above, by using simple transformations and substitutions we got the solution for the given question.
Hence, the given question is solved.

Note:
 We should be very careful while doing the calculation of the given question. We should be well aware of the transformations and substitutions that have to be made for the given question to get the equation more simplified. Once the equation gets simplified, we should do the calculation very carefully to get the given question solved. We should first understand the question correctly and then have to decide which transformation is to be used to get the question solved. It’s better for us to verify the value of $ x$ that we get like by substituting the value in the equation and seeing if right hand side matches with the left hand side or not mathematically as in the above case \[\dfrac{1}{2}\left( 4\left( 11 \right)+6 \right)=25\] and \[\dfrac{1}{3}\left( 9\left( 11 \right)-24 \right)=25\] .