Graphs of Absolute Value Equations
Almost all absolute value equations have a "V" shape. The absolute value \begin{align*}y=x\end{align*}
All absolute value equations are essentially variations of this "V" shape. The location and the slope of the "V" change depending on the equation. If the absolute value equation has a yintercept at the end of it, you must implement the yintercept when graphing the "V" shape. All you have to do is shift the V up by whatever number the yintercept is.
Absolute value equations will often have other numbers/terms inside the absolute value brackets that you need to consider when graphing an absolute value equation. If you have the equation \begin{align*}y=x1\end{align*}
It is important to know how to integrate the two rules stated above. Practice integrating the rules above by graphing the following absolute value equations:

\begin{align*}y=x4+7\end{align*}
y=x−4+7 
\begin{align*}y=x+93\end{align*}
y=x+9−3 
\begin{align*}y=x+32\end{align*}
y=x+3−2
Also know that the slopes in these types of problems won't always be 1. If there is a coefficient symbolizing the slope in an absolute value equation, be sure to change the slant of the "V" accordingly. Say your slope is x. On the right side of the "V", make your slope x. However, on the left side of the "V", your slope will be negative x.
\begin{align*}y=4x+2\end{align*}
Practice integrating graphing absolute value equations with more complicated slopes.

\begin{align*}y=4x2\end{align*}
y=4x−2 
\begin{align*}y=\left \frac{1}{2} x\right +6\end{align*}
y=∣∣∣12x∣∣∣+6
Finally, note that sometimes you will have to graph an upsidedown "V" shape. This will happen if there is a negative sign right before the absolute value term. For example, in the absolute value equation\begin{align*}x2=y\end{align*}
TIP: Explain why the following two graphs are shaped the way they are to a family member or friend. If you can successfully explain why to somebody else, you have a strong understanding yourself.
\begin{align*}x2=y\end{align*}
\begin{align*}y=x2\end{align*}
Overall Practice
Use all the principles described above to graph the following graphs.

\begin{align*}y=x+3\end{align*}
y=x+3 
\begin{align*}y=x6\end{align*}
y=x−6 
\begin{align*}y=4x+2\end{align*}
y=4x+2 
\begin{align*}y=\left \frac{x}{3}4\right \end{align*}
y=∣∣x3−4∣∣ 
\begin{align*}x4=y\end{align*}
x−4=y 
\begin{align*}x2=y\end{align*}
−x−2=y 
\begin{align*}y=x2\end{align*}
y=x−2 
\begin{align*}y=x+3\end{align*}
y=x+3 
\begin{align*}y=\frac{1}{2} x\end{align*}
y=12x 
\begin{align*}y=4x2\end{align*}
y=4x−2 
\begin{align*}y=\left \frac{1}{2} x\right +6\end{align*}
y=∣∣∣12x∣∣∣+6
Graphs of Inequalities in One & Two Variables
Review of graphing inequalities in two variables:
Graphing inequalities is fairly simple as long as you know how to graph lines. First graph the line that you are given changing the inequality sign to an equal sign for now. Once you have graphed your line (this applies to two variables as well as one variable), you will notice that you have divided the Cartesian plane into two pieces/halves. Inequality graphs require that you shade in one half/piece that you have created. To figure out which half you need to shade, pick an arbitrary point in one of the halfs. Plug the point into you picked into the inequality (with the equals sign changed back into the greater than/less than sign). Then see if the inequality is still true. If it is, shade in the section of the graph where the point you arbitrarily picked was in. If not, shade in the other side.
Review of graphing inequalities in one variable
Now that you know how to graph inequalities with two variables, graphing inequalities with one variable is extremely simple. If you had to graph the inequality \begin{align*}x>6\end{align*}
If you are graphing a simple absolute value inequality, all you have to do is rewrite the inequality with 2 "caps" that will define where you start and stop shading in. For the absolute value inequality \begin{align*}y < 5\end{align*}
\begin{align*}y > 5 \quad \text{and} \quad y < 5\end{align*}
Now all you have to do is shade in the area between your two caps, in this case between the y values of 5 and 5.