Absolute value of a function or number is that number of units of distance from the zero mark when drawn on a number line. It could either be left to the zero or right to the zero. The symbol used to denote absolute value is bars around the number or variable. Such that, $|x|$ is read as absolute value of $x$. It is $x$ units away from zero on either side on the number line. Inequality means lack of equality. It could either be less than or more than or not equal to. The symbols used to denote inequality are $<,\ >$ or $\neq$. So, combining the terms ‘absolute value’ and ‘inequality’ we can say that absolute value inequality is a type of inequality that possesses absolute value.

## Definition

An absolute value inequality is defined as an inequality that has modulus sign of absolute value containing a variable inside the bars. We need to consider two cases while solving absolute value inequalities. The cases are as follows:

• Case 1 - The equation written inside the modulus symbol of the absolute value is positive

• Case 2 - The equation written inside the modulus symbol of the absolute value is negative

The solution to absolute value problem would be the intersection of solution when each of the case is solved, such that if $|a| < b$, then $a < b$ AND $a > -b$

Absolute value inequality is the measure of the distance of any quantity from the origin $0$, such that $|x|$ measures the distance of $x$ from $0$. The fields in which absolute value inequality finds its application are symmetric, symmetric limits or boundary conditions.

## Double Absolute Values Inequalities

Double absolute value inequality can be illustrated with the help of an example taken like

$|x + 4|\ >\ |x – 2|,\ x\ E\ R$

The first step is to write the absolute inequality function as a combination of linear functions and separately solve the inequalities. It can be written as,

$|x + 4|$ = $\left\{\begin{matrix}x + 4,\ x + 4\ \geq 0,\ x\ \geq -4\\ -(x + 4),\ x < -4 \end{matrix}\right.$

$|x + 2|$ = $\left\{\begin{matrix}x - 2,\ x - 2\ >\ 0,\ x\ >\ 2\\ -(x - 2),\ x < 2\end{matrix}\right.$

In this way the absolute value function can be rewritten. We need to look into the $x$ values lying at different intervals. From the values of $x$ the three different interval of $x$ we get are as follows:
a) $(- \infty,\ -4)$

Both negative is considered

$-(x + 4)\ >\ -(x – 2)$

$-x\ -4\ >\ -x\ +\ 2$

$-4\ >\ 2$ No solution
b) $(-4,\ 2)$

$X + 4\ >\ -(x – 2)$

$X + 4\ >\ -x + 2$

$2x\ >\ -2$

$X\ >\ -1$
c)  $(2,\ \infty)$

$X + 4\ >\ x – 2$

$4\ >\ -2$

Always true

$[2,\ \infty)$

So, the answer is $x\ >\ -1$

## Solving Absolute Values Inequalities

Absolute value inequalities can be solved by following the below mentioned steps:-

Step 1: Absolute value inequality takes up a number of forms which need to be evaluated. The various forms of absolute value inequalities could be $|| <$ or $|| >$; $|+-|<$ or $|+-|>$

Step 2: An absolute value inequality is transformed into normal inequalities. The sign inside the bar could either be positive or negative. So, there are two cases which we need to consider changing into normal inequalities. They are if $|x| < a$, then $x < a$ or $x> -a$. The word used ‘or’ symbolizes that either of the two cases shall satisfy the given absolute value inequality problem.

Step 3: After we have broken the absolute value inequality into as many normal inequalities as possible, we replace the inequality sign with equality sign till the last step is reached.

Step 4: For addition or subtraction on either side it would be same as any equality problem but in case of multiplication or division on both sides the sign lying in the middle juxtaposes, which means lesser than symbol changes to greater than and vice versa.

Step 5: After solving for $x$ we write down the solution set. The solution set is actually the intersection of each of the normal inequalities being solved. It is the range of values that could be substituted as $x$ in the given absolute value inequality problem.

Step 6: Verify the answer. We choose a value of $x$ lying in the solution set and substitute it as value of $x$ in the given absolute inequality problem. If it satisfies the equation then its correct else the work needs to be checked.

## Graphing Absolute Values Inequalities

Absolute value inequalities can be graphed considering the two conditions. The first condition being $|x| < a$, which is if the absolute value of the variable $x$ is less than a then the resulting graph will be the portion between the two points $–a$ and $a$ drawn on the number line, with an open circle around $–a$ and $a$ as because those two values are excluded. The second condition being, if $|x| > a$, which is if the absolute value of the variable $x$ is more than $a$, then the resulting graph will be two rays heading towards infinity each starting from the point $–a$ to $– \infty$ and the other a to $+ \infty$ drawn on the number line with an open circle around $–a$ and $a$ as those two values are not considered. Any point lying on the portion between the two points or on the rays shall satisfy the given absolute value inequality problem.

## Examples

Example 1:

Solve $|2x - 6|\ >\ 8$

Solution:

The two cases are

$2x - 6\ >\ 8$

$2x\ >\ 14$

$X\ >\ 7$

And

$2x - 6\ <\ -8$

$2x\ <\ -2$

$X\ <\ -1$

Representing the solution on the number line.

Example 2:

A jam manufacturer restricts the amount of juice in each jar to have a net weight of $20$ ounces, with a tolerance of $1.5$ ounce. Write and solve an absolute value inequality that describes the acceptable net weights for jam in the jars

Solution:

The absolute value inequality that could be constructed based on the information given is

$|x - 20|\ \leq\ 1.5$

So, $x - 20\ \leq\ 1.5$ or $x - 20\ >\ -1.5$

Solving, we get $x\ \leq\ 21.5$ and $x\ >\ 18.5$

Thus, the acceptable net weight should range from $18.5$ ounces to $21.5$ ounces inclusive.