Squaring or canceling can create false solutions, so check the original equation

Absolute value can give two inputs for one output

For $|2x-5|=9$, graph $y=|2x-5|$ and $y=9$. The crossings have inputs $-2$ and $7$. The two algebraic cases are $2x-5=-9$ and $2x-5=9$.

Absolute value is never negative. An equation such as $|2x-5|=-9$ has no real solution, which you can establish before opening the calculator.

Squaring can introduce an invalid solution

For $\sqrt{2x+11}=x-2$, the right side must be nonnegative, so any solution must have $x\ge2$. Squaring gives $x^2-6x-7=0$, whose candidates are $7$ and $-1$.

Check the original equation: $x=7$ gives $5=5$, while $x=-1$ gives $3=-3$. Only $7$ works. Graphing the original two sides avoids treating every solution of the squared equation as valid.

$\displaystyle y=\sqrt{2x+11}$

The nonnegative square-root output.

$\displaystyle y=x-2$

The other side of the original equation.

A canceled factor still makes the original denominator zero

For $(x^2-9)/(x-3)=6$, factoring suggests $x+3=6$, hence $x=3$. But the original denominator is zero there. The original equation has no solution.

Never count a jump across an asymptote as an intersection. Zoom, evaluate both original sides, and check the denominator. The same habits protect you from tiny holes that a screen cannot show clearly.

Keep practicing the connected skills