Solution of Equations and Inequalities

Solve: $\frac{x}{x+2}+\frac{2}{\left(x+2\right)\left(x+3\right)}=\frac{5}{x+3}$.

Solve the equation: $2x+\frac{6}{x}=7$

Solve $|x-3|=-8$ using modulus function definition.

Solve $|x+3|=8$ using modulus function definition.

Solve: 

$3+\sqrt{3x+1}=x$

Solve: 

$x=\sqrt{3x}+6$

If the extraneous root or solution for the equation $\sqrt{x-1}+4=x-3$ is $x=k$, then find the value of $k$.

Find the extraneous root or solution for the equation $4+\sqrt{x+2}=x$.

Solve the following inequality and write the solution in interval notation:

$\frac{5}{x^2-2x-15}>0$