Question

How do you graph and solve $|3x-12| > 0$?

Answer 1

We will start by analyzing the given equation. We will look at the definition of absolute value and then its meaning with respect to the given equation. We are given an inequality, so we will interpret the meaning of the inequality. We will look at the graph representing the given equation. Then we will determine the solution of the inequality.

Complete step by step answer:
The absolute value is also called the modulus function. It is written as $f\left( x \right)=|x|$. The absolute value of a number or an expression is the non-negative value of that number or expression. That means, if $x\ge 0$, then $|x|=x$ and if $x < 0$, then $|x|=-x$.
The given expression is $f\left( x \right)=|3x-12|$. We are considering the absolute value of the expression. So, according to the definition of the absolute value, we have to look at the non-negative value of the given expression.
In the question, we are given an inequality to the expression. The given inequality is $|3x-12| > 0$. This inequality tells us to consider all the values of the expression that are greater than 0.
The graph of $f\left( x \right)=|3x-12|$ looks like the following,

We can see that at $x=4$, we get $f\left( x \right)=0$. We can see that $3x-12 > 0\text{ if }x > 4$ and the shaded region in the following graph represents this inequality,
Also, we have $3x-12 < 0\text{ if }x < 4$. The graph for this case looks like the following,

So, the given inequality represents the union of both the shaded regions from the above two graphs.

Note: While solving inequalities, it is good to represent the regions of the inequalities in a distinct way. This allows us to keep track of the regions for different inequalities and helps us in avoiding confusions. The modulus function is an important function. There are many identities which make use of the absolute value or modulus to showcase the properties of the objects involved in the identities.