1) Write \(\displaystyle -6 \le 6x + 3 \le 21\) as two simple inequalities:

   \(\displaystyle -6 \leq 6x + 3\)          \(\displaystyle 6x + 3 \leq 21\)

2) Solve the inequalities:

   \(\displaystyle -3 - 6 \leq 6x\)            \(\displaystyle 6x \leq 21 - 3\)

   \(\displaystyle -9 \leq 6x\)                   \(\displaystyle 6x \leq 18\)

   \(\displaystyle -\frac{9}{6} \leq x\)                    \(\displaystyle x \leq 3\)

3) Write the final solution as a single compound inequality:

   \(\displaystyle -\frac{9}{6} \leq x \leq 3\)

 For interval notation:

 \(\displaystyle [-\frac{9}{6}, 3] = [-1 \frac{3}{6}, 3]\)

4) Now graph:

In order to solve this equation, we must first isolate the absolute value. In this case, we do it by dividing both sides by \(\displaystyle 3\) which leaves us with:

\(\displaystyle \left | x-6\right |< 7\)\(\displaystyle \left | x-6 \right|< 7\)

When we work with absolute value equations, we're actually solving two equations. So, our next step is to set up these two equations: 

\(\displaystyle x-6< 7\) and \(\displaystyle x-6>-7\)

In both cases we solve for \(\displaystyle x\) by adding \(\displaystyle 6\) to both sides, leaving us with

\(\displaystyle x< 13\) and \(\displaystyle x>-1\)

This can be rewritten as \(\displaystyle -1< x< 13\)

When we work with absolute value equations, we're actually solving two equations: 

\(\displaystyle 4x-2< 10\) and \(\displaystyle 4x-2>-10\)

Adding \(\displaystyle 2\) to both sides leaves us with: 

\(\displaystyle 4x< 12\) and \(\displaystyle 4x>-8\)

Dividing by \(\displaystyle 4\) in order to solve for \(\displaystyle x\) allows us to reach our solution:

\(\displaystyle x< 3\) and \(\displaystyle x>-2\)

Which can be rewritten as:

\(\displaystyle -2< x< 3\)

In order to solve for \(\displaystyle x\) we must first isolate the absolute value. In this case, we do it by dividing both sides by 2:

\(\displaystyle \left | 3x-5\right |>23\)\(\displaystyle |3x-5|>23\)

As with every absolute value problem, we set up our two equations:

\(\displaystyle 3x-5>23\) and \(\displaystyle 3x-5< -23\)

We isolate \(\displaystyle x\) by adding \(\displaystyle 5\) to both sides:

\(\displaystyle 3x>28\) and \(\displaystyle 3x< -18\)

Finally, we divide by \(\displaystyle 3\):

\(\displaystyle x>\frac{28}{3}\) and \(\displaystyle x< -6\)

Our first step in solving this equation is to isolate the absolute value. We do this by dividing both sides by \(\displaystyle 5:\)

\(\displaystyle |\frac{x}{2}+4|< 2\).

We then set up our two equations:

\(\displaystyle \frac{x}{2}+4< 2\)\(\displaystyle \frac{x}{2}+4< 2\) and \(\displaystyle \frac{x}{2}+4>-2\)\(\displaystyle \frac{x}{2}+4>-2\).

Subtracting 4 from both sides leaves us with

\(\displaystyle \frac{x}/{2} < -2\)\(\displaystyle \frac{x}{2}< -2\) and \(\displaystyle \frac{x}{2}>-6\)\(\displaystyle \frac{x}{2} >-6\).

Lastly, we multiply both sides by 2, leaving us with \(\displaystyle x\):

\(\displaystyle x< -4\) and \(\displaystyle x>-12\).

Which can be rewritten as:

\(\displaystyle -12< x < -4\)

We first need to isolate the absolute value, which we can do in two steps:

1. Add 2 to both sides:

\(\displaystyle 4\left | 5x+8\right |< 24\)

2. Divide both sides by 4:

\(\displaystyle \left | 5x+8\right |< 6\)\(\displaystyle |5x+8|< 6\)

Our next step is to set up our two equations:

\(\displaystyle 5x+8< 6\) and \(\displaystyle 5x+8>-6\)

We can now solve the equations for \(\displaystyle x\) by subtracting both sides by 8:

\(\displaystyle 5x< -2\) and \(\displaystyle 5x>-14\)

and then dividing them by 5:

\(\displaystyle x< -\frac{2}{5}\) and \(\displaystyle x>-\frac{14}{5}\)

Which can be rewritten as: 

\(\displaystyle -\frac{14}{5}< x< -\frac{2}{5}\)

\(\displaystyle 7+\left | 4x+13\right |< 36\)

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by subtracting seven from both sides.

\(\displaystyle \left | 4x+13\right |< 29\)

Next we need to set up two inequalities since the absolute value sign will make both a negative value and a positive value positive.

\(\displaystyle -29< 4x+13< 29\)

From here, subtract thirteen from both sides and then divide everything by four.

\(\displaystyle -29-13< 4x< 29-13\)

\(\displaystyle -42< 4x< 16\)

\(\displaystyle -10.5< x< 4\)

\(\displaystyle 3\left | \frac{2x}{5} +7\right |>18\)

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by dividing both sides by three. 

\(\displaystyle \left | \frac{2x}{5} +7\right |>6\)

We now have two equations:

\(\displaystyle \frac{2x}{5} +7>6\)  and \(\displaystyle -6> \frac{2x}{5} +7\)

\(\displaystyle \frac{2x}{5} >-1\)               \(\displaystyle -13> \frac{2x}{5}\)

\(\displaystyle x>-\frac{5}{2}\)                  \(\displaystyle -\frac{65}{2}> x\)

So, our solution is \(\displaystyle -32.5>x \ or -2.5< x\)

\(\displaystyle 2+3\left | x+4\right |< 17\)

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by subtracting two from both sides then dividing everything by three.

\(\displaystyle 3\left | x+4\right |< 15\)

\(\displaystyle \left | x+4\right |< 5\)

Since absolute value signs make both negative and positive values positive we need to set up a double inequality.

\(\displaystyle -5< x+4< 5\)

Now to solve for \(\displaystyle x\) subtract four from each side.

\(\displaystyle -9< x< 1\)

If \(\displaystyle \left | 5-2x \right |>11\), then either \(\displaystyle 5-2x > 11\) or \(\displaystyle 5-2x < -11\) based on the meaning of the absolute value function. We have to solve for both cases.

a) \(\displaystyle 5-2x > 11\) subtract 5 from both sides

\(\displaystyle -2x > 6\) divide by -2, which will flip the direction of the inequality

\(\displaystyle x < -3\) 

Even if we didn't know the rule about flipping the inequality, this answer makes sense - for example, \(\displaystyle 1 < 3\), and \(\displaystyle -2(-4) =8 > 6\).

b) \(\displaystyle 5-2x < -11\) subtract 5 from both sides

\(\displaystyle -2x < -16\) divide by -2, once again flipping the direction of the inequality

\(\displaystyle x > 8\)