Vocab: An absolute value equation is... You guessed it!... an equation with absolute values.
Ahhh... A Super Simple Example
Solve the absolute value equation $|x|=5$.
In General
To solve an absolute value equation of the form $|\mbox{Stuff}|=n$, solve the two equations $$\mbox{Stuff}=n$$ and $$\mbox{Stuff}=-n$$ Example: Solve the absolute value equation $|3(x-5)+4(-4x-2)|=3$.
Another Form
To solve an absolute value equation of the form $|\mbox{Stuff}_1|=|\mbox{Stuff}_2|$, solve the two equations $$\mbox{Stuff}_1=\mbox{Stuff}_2$$ and $$\mbox{Stuff}_1=-\mbox{Stuff}_2$$ Example: Solve the absolute value equation $|-5x+2|=|-x-3|$.
Absolute Value Inequalities
Ahhh. Another Easy Example: Solve the absolute inequality $|x|<5$.
Also, graph the solution set and write it in interval notation.
In General
To solve an absolute value inequality of the form $|\mbox{Stuff}| \lt n$, solve the compound inequality $$-n \lt \mbox{Stuff} \lt n$$ To solve an absolute value inequality of the form $|\mbox{Stuff}| \leq n$, solve the compound inequality $$-n \leq \mbox{Stuff} \leq n$$ Example: Solve the absolute value inequality $|3x+1| \leq 4$.
One More Easy Example: Solve the absolute inequality $|x|>5$.
Also, graph the solution set and write it in interval notation.
In General
To solve an absolute value inequality of the form $|\mbox{Stuff}| \gt n$, solve the compound inequality $$\mbox{Stuff} \lt -n \,\,\,\mbox{ OR }\,\,\, n \lt \mbox{Stuff}$$ To solve an absolute value inequality of the form $|\mbox{Stuff}| \geq n$, solve the compound inequality $$\mbox{Stuff} \leq -n \,\,\,\mbox{ OR }\,\,\, n \leq \mbox{Stuff} $$ Example: Solve the absolute value inequality $|-4(2x+3)-2(-3x-1)| \gt 1$.
Application
Tolerance of a Soft Drink Bottle The specifications for a bottle-filling machine in a soft drink plant call for the machine to dispense 16 oz into a bottle with a tolerance of $\pm$0.25 oz. Express this tolerance interval as an absolute value inequality.