When taking absolute values, we need to consider both positive and negative values. So, we have two equations. $\displaystyle -(x+1)=3, x+1=3$ 

For the left equation, we can switch the minus sign to the other side to get $\displaystyle x+1=-3$. When we subtract $\displaystyle 1$ on both sides, we get $\displaystyle x=-4$.

For the right equation, just subtract $\displaystyle 1$ on both sides, we get $\displaystyle x=2$.

$\displaystyle x=2, x=-4$

When taking absolute values, we need to consider both positive and negative values. So, we have two equations. $\displaystyle 4x=12, -(4x)=12$ 

For the left equation, when we divide both sides by $\displaystyle 4$, $\displaystyle x=3$. 

For the right equation, we distribute the negative sign to get $\displaystyle -4x=12$. When we divide both sides by $\displaystyle -4$, $\displaystyle x=-3$. 

$\displaystyle x=3, x=-3$

When taking absolute values, we need to consider both positive and negative values. Let's first subtract $\displaystyle 5$ on both sides. So, we have two equations. $\displaystyle 2x=4, -(2x)=4$ 

For the left equation, when we divide both sides by $\displaystyle 2$, $\displaystyle x=2$. 

For the right equation, we distribute the negative sign to get $\displaystyle -2x=4$. When we divide both sides by $\displaystyle -2$, $\displaystyle x=-2$. 

$\displaystyle x=2, x=-2$

When taking absolute values, we need to consider both positive and negative values. So, we have two equations. $\displaystyle 16+4x=4+2x, 16+4x=4-2x$ 

For the left equation, we subtract $\displaystyle 4$ on both sides and subtract $\displaystyle 4x$ on both sides. We now have $\displaystyle 12=-2x$. When we divide both sides by $\displaystyle -2$, $\displaystyle x=-6$. 

For the right equation, we subtract $\displaystyle 4$ on both sides and subtract $\displaystyle 4x$ on both sides. We now have $\displaystyle 12=-6x$. When we divide both sides by $\displaystyle -6$, $\displaystyle x=-2$. 

Let's double check. When we plug in $\displaystyle -6$, both sides aren't equal.

$\displaystyle 16+4(-6)=4+2\left | -6\right |$

$\displaystyle 16-24=4+12$

$\displaystyle -8\neq16$

But if we plug in $\displaystyle -2,$ we get both sides equal.

$\displaystyle 16+4(-2)=4+2\left | -2\right |$

$\displaystyle 16-8=4+4$

$\displaystyle 8=8$

So $\displaystyle -2$ is the only answer. 

Let's isolate the variable by subtracting both sides by $\displaystyle 3$. We have:

$\displaystyle -\left | x\right |=2$ This will be a contradicting expression. Absolute values always generate positive values and since there's a negatie sign in front of it, it will never match a positive value. Therefore no possible answer exist. 

When taking absolute values, we need to consider both positive and negative values. So, we have two equations. $\displaystyle -x-6=0, x-6=0$ 

For the left side, we add $\displaystyle 6$ to both sides and shift the negative sign to the other side to get $\displaystyle x=-6$.

For the right side, we add $\displaystyle 6$ to both sides and $\displaystyle x=6$.

$\displaystyle x=6, x=-6$

Let's isolate the variable by subtracting both sides by $\displaystyle 6$. We have:

$\displaystyle \left | x\right |=-6$ This will be a contradicting expression. Absolute values always generate positive values. Therefore no possible answer exist. 

When taking absolute values, we need to consider both positive and negative values. Let's multiply both sides by $\displaystyle \left | x\right |$ to get rid of the fraction. So, we have two equations. $\displaystyle 9x=6, -(9x)=6$ 

For the left equation, when we divide both sides by $\displaystyle 9$, $\displaystyle x=\frac{2}{3}$. 

For the right equation, we distribute the negative sign to get $\displaystyle -9x=6$. When we divide both sides by $\displaystyle -9$, $\displaystyle x=-\frac{2}{3}$. 

$\displaystyle x=\frac{2}{3}, x=-\frac{2}{3}$

When taking absolute values, we need to consider both positive and negative values. Let's multiply each side by $\displaystyle 3$ to get rid of the fraction. So, we have two equations. $\displaystyle 4x=24, -(4x)=24$ 

For the left equation, when we divide both sides by $\displaystyle 4$, $\displaystyle x=6$. 

For the right equation, we distribute the negative sign to get $\displaystyle -4x=24$. When we divide both sides by $\displaystyle -4$, $\displaystyle x=-6$. 

$\displaystyle x=6, x=-6$

When taking absolute values, we need to consider both positive and negative values. So, we have two answers. $\displaystyle x=17, x=-17$