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 $x \le \frac C A$,

When $B \ne 0$ and $ A>0 $ the inequality can be solved so that $x \le \frac {C- B} A$,

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 $x$.

Here is another example showing that important connection:

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