Linear inequality in one variable

A linear inequality in one variable is of the form:

$ax+b\lt c$

$ax+b\le c$

$ax+b\gt c$

or

$ax+b\ge c$

where $x$ is a variable, $a$,$b$ and $c$ are constants, i.e., some numbers.

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Addition and subtraction properties of inequality

If $a\lt b$ then

$a+c\lt b+c$

and

$a-c\lt b-c$

Multiplication and division properties of inequality

For any real numbers $a, b$ and $c$,

If $a\lt b$ then

$ac\lt bc$

and

$\dfrac{a}{c}\lt \dfrac{b}{c}$ if $c$ is positive

$\dfrac{a}{c}\gt \dfrac{b}{c}$ if $c$ is negative

Compound inequalities

If we have two or more sets, we can combine them with union and intersection operations.

If $A$ and $B$ are two sets, then the union of $A$ and $B$ is the set of elements which are in $A$ or in $B$ or in both $A$ and $B$.

Union of $A$ and $B$ or $A$ union $B$ is denoted as $A\cup B$

If $A$ and $B$ are two sets, then the intersection of $A$ and $B$ is the set of elements which are common to both $A$ and $B$.

Intersection of $A$ and $B$ or $A$ intersection $B$ is denoted as $A\cap B$

A B

$A\cup B$

A B

Absolute value equation

Absolute value of a number is the distance of the number from zero on the number line. Since distance is always positive, absolute value is positive. Absolute value of a number, $A$ is written as $|A|$.

Solving absolute value equation

If $|x|=a$ then $x=a$ or $x=-b$ if $a$ is positive or zero.

If $|x|=a$ then there is no solution if $a$ is negative.

Solving absolute value inequalities

For any positive real number, $a$, i.e., $a>0$,

If $|x|\gt a$, then $x\lt -a\:$ or $\:x\gt a$

If $|x|\lt a$, then $-a\lt x\lt a$

Solving inequalities- test point method

There are four steps in this method.
  1. Find the boundary points of the inequality. To find that replace the inequality sign with the equal sign and find the solutions. The solutions are the boundary points.
  2. Plot the boundary points on a number line. You will have different intervals in the number line.
  3. Pick a test point from each interval and substitute it into the inequality. If the test point makes the inequality true, then