Consider the linear function $f(x) = mx + b$. 

A standard form of a linear inequality with variable $x$ is an inequality of the form $Ax + B \le C$ with $A \ne 0$.
Beginning algebra spends a considerable amount of time solving this inequality without any reference to functions.
When $B = 0$ and  $ A>0 $ this inequality has the solution $x \le \frac C A$, i.e., $x \in (-\infty, \frac C A]$.
When $B \ne 0$ and  $ A>0 $ the inequality can be solved so that $x \le  \frac {C- B} A$, i.e., $x \in (-\infty, \frac {C-A} A]$.

Example  LInEQ.1 : Suppose $5x - 10 \le 20$. Solve the inequality for $x$.
A slightly more ambitious form of a linear inequality with variable $x$ is an inequality of the form $A_1x + B \le A_2x +C$ with $A_1 \ne A_2$. Algebraically this is solved by solving the related inequality $Ax + B \le C$ where $A = A_1 - A_2$. For example the inequality $8x - 10 \le 3x + 20$ would be solved by solving the inequality $5x - 10 \le 20$ as in Example LInEQ.1.

As demonstrated in Example LInEQ.1, there is an important connection between the algebraic steps for solving a linear inequality and a mapping diagram for a linear function understood as a composition.
Here is another example showing that important connection:
Example LInEQ.2 : Suppose $2x + 1 \le 2$. Solve the inequality for $x$.

You can use this next dynamic example to solve linear inequalities like those in Examples LInEQ.1 and LInEQ.2.


Example LF.DLInEQ.0 Dynamic Views for solving an inequality $f(x) = mx+b \le c$ on  Mapping Diagrams