IX Linear Equations and Linear Inequalities

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.

$8 \geq x + 2 > 0$ | $-2$
$6 \geq x > -2$
$(-2,6]$