#### Answer

$x=12$

#### Work Step by Step

Square both sides of the equation to obtain:
$2x+1=(x-7)^2$
Use the rule $(a-b)^2 = a^2 - 2ab+b^2$ where $a=x$ and $b=7$ to obtain:
$\\2x+1=x^2-14x+49$
Move all terms to the right side of the equation. Note that when a term is moved to the other side of the equation, its sign changes to its opposite.
$0=x^2-14x+49-2x-1
\\0=x^2-16x+48$
Factor the trinomial to obtain:
$0=(x-4)(x-12)$
Equate each factor to zero then solve each equation to obtain:
$x -4 = 0 \text{ or } x - 12 = 0
\\x = 4 \text{ or } x = 12$
Note that when $x=4$,
$\sqrt{2(4)+1}= 4-7
\\\sqrt{8+1} = -3$,
Since the principal square root cannot be negative, this means that $4$ is an extraneous solution.
Thus, the solution is $x=12$.