## Linear Equations

In addition to possessing one solution linear equations may also possess *zero* or *infinite* solutions.

Example of linear equation with no solution:

$4(x+2)-2(2x+3)=10$

$4x+8-4x-6=10$

$2=10$ | results in a false statement

Example of linear equation with infinite solutions:
$2x-(x+5)=x-5$

$2x-x-5=x-5$

$x-5=x-5$

$-5=-5$ | results in a true statement for any value of x

## Linear Inequalities

### Interval Notation

Interval notation describes a starting point and an ending point of a number range. Numbers next to parantheses are excluded from the number range while numbers next to brackets are included in the number range described by the notation.

$x < 3$ is $(-\infin, 3)$

$x > -2$ is $(-2, \infin)$

$x \leq -5$ is $(-\infin, -5]$

$x \geq 4$ is $[4, \infin)$

### Solving Inequalities

*Inequalities* are for the most part solved the exact same was as *equations*, however after **dividing or mulitplying** by a **negative number** the *inequality* has to be *reversed* (greater than becomes lesser than and vice versa).

$-5x + 4 \leq 44$ | $-4$

$-5x \leq 40$ | $\div -5$

$x \geq -8$

$[-8,\infin)$

### Compound Inequalities

Multiple inequalities chained together with **AND** or **OR** conjunctions are called *compound inequalities*. If both inequalities chained by an **AND** conjuction contradict each other the result is an *empty set of numbers* denoted by the symbol $\text{\O}$.

### Double Inequalities

These define the numbers inbeweteen limits. Example: $5 > x \geq 2$ is $[2,5)$. Operations have to be attributed to all parts of double inequalities when solving them.

$8 \geq x + 2 > 0$ | $-2$

$6 \geq x > -2$

$(-2,6]$