Solve Inequalities

Inequalities are like equations.  You solve them the same way you solve equations.  However, at the end, you do not find the exact value of a variable.  Instead you learn the parameters of a value, what it's greater than or less than.

The type of equality you're solving is deteremined by the operating symbol that replaces the equal sign.  There are four inequality symbols.

Think about this example:

$x + 5 \leq 9$

What do you know about $x$?

Use the same strategies you use in equations to isolate the variable:

$\eqalign{x + 5 \leq 9\\ -5 \quad -5\\ x \leq 4}$

$x$ is less than or equal to $4$

That's how you solve an inequality.  If the process does not look entirely familiar, review the Solving for a Variable lesson. If the know the process of solving for a variable well, you just need to know a few special rules for solving inequalities.  

When you multiply each side of an inequality by a negative number, flip the inequality sign.

When you dividide each side of an inequality by a negative number, flip the inequality sign.

You may flip the inequality sign several times over the course of a problem!

Why flip the sign?  Think about a number line.  When positive numbers get more positive, they move to the right.  When negative numbers get more negative, they move to the left.  So, if you are thinking about a number getting more positive or more negative, it's moving in a different direction, depending on what side of the 0 it starts on.  So, when you flip from positive to negative, or vice versa, you also flip the inequality sign. 
 

Let's try an example:

$$\eqalign{-2x + 5 &> 12 &&\text{We want to isolate the x.}\\-5 & \quad -5 &&\text{Subtract 5 from both sides}\\-2x&>7&&\text{Divide both sides by -2 (and flip the sign!)}\\\div -2 & \quad \div -2\\x&<-3.5&&\text{You have isolated your variable.  x is less than -3.5}}$$

So, how do you graph an inequality?

These are linear inequalities so they are one-dimensional.

When you graph a linear inequality, you draw a ray.   "Greater than" rays point to the right.  "Less than" rays point to the left.  At the stopping point of the ray, you put either an open circle (if the inequality is "greater than" or "less than") or a closed circle (if the inequality is "greater than or equal to" or "less than or equal to").

Why a different circles?  A solid circle INCLUDES the number marked by the circle (a variable that is "greater than or equal to 3" could be any number equal to or greater than 3, including 3, thus a closed circle over 3).  An open circle includes every number greater than or less than the number marked by the circle, but NOT the number (so a variable greater than 3 includes every number greater than 3, but not 3).

Let's see some examples:

$x > 3$  graphs as:   $\dfrac{\circ \!\! \rightarrow}{3 \quad \,}$

$x \geq 3$  graphs as:   $\dfrac{\bullet \!\! \rightarrow}{3 \quad \,}$

$x < 3$  graphs as:   $\dfrac{\leftarrow \!\!\circ}{\, \quad 3 }$

$x \leq 3$  graphs as:   $\dfrac{\leftarrow \!\! \bullet }{\, \quad 3}$