Answer
$x=-\dfrac{60}{37}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\dfrac{2}{5}x-\dfrac{3}{2}x=\dfrac{3}{4}x+3
,$ remove first the fraction by multiplying both sides by the $LCD.$ Then use the properties of equality to isolate the variable. Do checking of the solution.
$\bf{\text{Solution Details:}}$
The $LCD$ of the denominators, $\{
5,2,4,1
\},$ is $
20
$ since this is the least number that can be evenly divided (no remainder) by all the denominators. Multiplying both sides by the $LCD,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{2}{5}x-\dfrac{3}{2}x=\dfrac{3}{4}x+3
\\\\
20\left( \dfrac{2}{5}x-\dfrac{3}{2}x \right) =20\left( \dfrac{3}{4}x+3 \right)
\\\\
8x-30x=15x+60
.\end{array}
Using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
8x-30x=15x+60
\\\\
8x-30x-15x=60
\\\\
-37x=60
\\\\
x=\dfrac{60}{-37}
\\\\
x=-\dfrac{60}{37}
.\end{array}
Checking: If $x=-\dfrac{60}{37},$ then
\begin{array}{l}\require{cancel}
\dfrac{2}{5}x-\dfrac{3}{2}x=\dfrac{3}{4}x+3
\\\\
\dfrac{2}{5}\left( -\dfrac{60}{37} \right) -\dfrac{3}{2} \left( -\dfrac{60}{37} \right)=\dfrac{3}{4}\left( -\dfrac{60}{37} \right)+3
\\\\
\dfrac{2}{\cancel5}\left( -\dfrac{\cancel5(12)}{37} \right) -\dfrac{3}{\cancel2} \left( -\dfrac{\cancel2(30)}{37} \right)=\dfrac{3}{\cancel4}\left( -\dfrac{\cancel4(15)}{37} \right)+3
\\\\
-\dfrac{24}{37}+\dfrac{90}{37}=-\dfrac{45}{37}+\dfrac{111}{37}
\\\\
\dfrac{66}{37}=\dfrac{66}{37}
\text{ (TRUE) }
.\end{array}
Hence, the solution is $
x=-\dfrac{60}{37}
.$
