Answer
$x=\dfrac{46}{5}$
Work Step by Step
For easier work, it is better to get rid of the fractions.
This can be achieved by multiplying the LCD of $21$ on both sides of the equation to obtain:
$21(\frac{x+1}{3}) = 21(5-\frac{x+2}{7})
\\\frac{21(x+1)}{3} = 21(5) - \frac{21(x+2)}{7}
\\7(x+1) = 105-3(x+2)
\\7x+7 = 105-3x-6
\\7x+7=-3x+99$
Add $3x$ and subtract $7$ on both sides to obtain:
$7x+3x=99-7
\\10x=92
\\\frac{10x}{10} = \frac{92}{10}
\\x = \frac{46}{5}$
Check:
$\begin{array}{ccc}
&\dfrac{\frac{46}{5}+1}{3} &= &5-\dfrac{\frac{46}{5}+2}{7}
\\&\dfrac{\frac{46}{5} + \frac{5}{5}}{3} &= &5-\dfrac{\frac{46}{5} + \frac{10}{5}}{7}
\\&\dfrac{\frac{51}{5}}{3} &= &5-\dfrac{\frac{56}{5}}{7}
\\&\frac{51}{5(3)}&= &5-\frac{56}{5(7)}
\\&\frac{17}{5} &= &5-\frac{8}{5}
\\&\frac{17}{5} &= &\frac{25}{5} -\frac{8}{5}
\\&\frac{17}{5} &= &\frac{17}{5}\end{array}$
