Answer
no solution
Work Step by Step
Using the properties of equality, the given equation, $
5|6-5x|=15x-35
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{5|6-5x|}{5}=\dfrac{15x-35}{5}
\\\\
|6-5x|=3x-7
.\end{array}
Removing the absolute value sign, the expression above is equivalent to
\begin{array}{l}\require{cancel}
6-5x=3x-7
\\\\\text{ OR }\\\\
6-5x=-(3x-7)
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
6-5x=3x-7
\\
-5x-3x=-7-6
\\
-8x=-13
\\
\dfrac{-8x}{-8}=\dfrac{-13}{-8}
\\
x=\dfrac{13}{8}
\\\\\text{ OR }\\\\
6-5x=-(3x-7)
\\
6-5x=-3x+7
\\
-5x+3x=7-6
\\
-2x=1
\\
\dfrac{-2x}{-2}=\dfrac{1}{-2}
\\
x=-\dfrac{1}{2}
.\end{array}
Since the right side of the original equation is not a constant, then checking of solutions is required.
Substituting $
x=\dfrac{13}{8}
$ in the original equation results to
\begin{array}{l}\require{cancel}
5|6-5x|=15x-35
\\\\
5\left| 6-5\left( \dfrac{13}{8} \right) \right|=15\left( \dfrac{13}{8} \right)-35
\\\\
5\left| 6-\dfrac{65}{8} \right|=\dfrac{195}{8}-35
\\\\
5\left| \dfrac{48}{8}-\dfrac{65}{8} \right|=\dfrac{195}{8}-\dfrac{280}{8}
\\\\
5\left| -\dfrac{17}{8} \right|=-\dfrac{85}{8}
\\\\
5\left(\dfrac{17}{8} \right)=-\dfrac{85}{8}
\\\\
\dfrac{85}{8}=-\dfrac{85}{8}
\text{ (FALSE)}
.\end{array}
Since the substitution above ended with a FALSE statement, then $
x=\dfrac{13}{8}
$ is not a solution.
Substituting $
x=-\dfrac{1}{2}
$ in the original equation results to
\begin{array}{l}\require{cancel}
5|6-5x|=15x-35
\\\\
5\left| 6-5\left( -\dfrac{1}{2} \right) \right|=15\left( -\dfrac{1}{2} \right)-35
\\\\
5\left| 6+\dfrac{5}{2} \right|=-\dfrac{15}{2}-35
\\\\
5\left| \dfrac{12}{2}+\dfrac{5}{2} \right|=-\dfrac{15}{2}-\dfrac{70}{2}
\\\\
5\left| \dfrac{17}{2} \right|=-\dfrac{85}{2}
\\\\
5\left(\dfrac{17}{2} \right)=-\dfrac{85}{2}
\\\\
\dfrac{85}{2}=-\dfrac{85}{2}
\text{ (FALSE)}
.\end{array}
Since the substitution above ended with a FALSE statement, then $
x=-\dfrac{1}{2}
$ is not a solution.
Hence, there is no solution.
