Absolute Value Inequalities

We solve inequalities as if they were an equation, except instead of an equal sign that is in the equation, there is a greater than sign $>$, or a less than sign $<$.
But there is another rule: When both sections are multiplied or divided by a negative number, the inequality sign is reversed.

When we are given an absolute value inequality, we do not know if the expression as an absolute value is positive or negative and therefore we will have to divide it into two cases.
The first case: The expression that is inside the absolute value is equal to $0$.
and the second case: The expression inside the absolute value is less than $0$.

Let's start with the question, What is absolute value?

An absolute value is the distance from the zero point.
What does that mean?
You can take any number in an absolute value and ask yourself: how far is it from $0$?
And that would be the correct answer.
Another important thing you should know, an absolute value is symbolized as follows: $││$
When you want to ask what the absolute value of a specific number is, place it inside the marking of these two bars.

Let's demonstrate.

If we want to know the absolute value of the number $5$ for example, we will write:
$│5│= ?$
We ask ourselves, how far is $5$ from $0$? And the answer is, of course, $5$.
In fact, when there is a positive number in an absolute value, the absolute value does not affect it and it will remain the same number just without an absolute value.
Therefore, an absolute value of $5$ is $5$.
$│5│=5$
Moreover, also fractions, non-integer numbers in absolute value, will remain exactly the same number.
Any positive number, unaffected by the absolute value.
We demonstrate:
$│2/3│= 2/3$

What happens when there is a negative number as an absolute value?
When there is a negative number inside the absolute value, the same question remains: What is its distance from point $0$?
The distance must be positive, there is no such thing as a negative distance. So, when there is a negative number in an absolute value, it will be equal to the same number but without the minus sign.
Let's demonstrate.

$│-5│=5$
In the same way:
$│-2/3│= 2/3$

In fact, we can say that the absolute value of a negative number is equal to the same positive number, both in absolute value and without it.
That is:
$│-5│= │5│= 5$
The distance from $5$ to $0$ and from $-5$ to $0$ is the same: $5$.
Therefore we can write:
$│-X│= │X│= X$

Now, we place the absolute value in an equation.
Example:
$│X│=4$
When we remember that an absolute value is equal to a distance from the zero point, we can ask ourselves, which $X$ is exactly four steps away from $0$?
There are $2$ answers.
$4$ is four steps away from $0$ and also $-4$ is four steps away from $0$.
Therefore:
$X=4, -4$
Now, let's move on to the next step of absolute value and use it in a slightly more complicated equation. Don't be scared, it's not much more complicated.
$│X+3│= 7$
How do you approach such an equation? Easily.
Simply replace the entire expression inside the absolute value with the word something.
$-7 =$ │something│
We ask ourselves, what something in absolute value can be equal to $7$? Of course!
$7$ or $-7$.

Now, we will replace instead of "something" what appears inside the absolute value, in the two equations we obtained.
That is:
In one case: $X+3=-7$
And in the second case: $X+3=7$

Magnificent! These are already equations that we can easily solve.
In one case: $X=-10$
And in the second case: $X=4$
Therefore, the solution to the exercise are the two $X$ obtained.
$X=-10 ,4$
Now, we move on to the second topic we want to present.

In the inequality, we obtain a result that is basically a domain of values. For example, $X$ is greater than some number or $X$ is less than some number.
How do we solve inequalities?
We solve inequalities exactly like an equation only that instead of an equal sign that is in the equation, there is a greater than sign $>$, or less than sign $<$
An important rule to keep in mind is the inequality: when you double or divide both sections by a negative number, the sign of the inequality is reversed.
For example:
$3X-5<X-11$
In the solution, as in the equation, we will transfer all the $X$ to one section and all the free numbers to the other segment.
We obtain:
$2X<-6$
Divide by $2$ both sections as in the equation and we obtain:
$X<-3$

An example where the inequality is reversed:
$3X+4<6X-11$
We will move the sections:
$-3X<-15$
Now we will divide both sections by: $-3$.
Since we divide by a negative number, we reverse the inequality.

That is:
$X>5$

Now, as promised, we will combine the two topics we have studied.

Geometric solution:
The geometric meaning of an absolute value is divided into two cases:
$│X│$ will be the distance between $X$ and $a$.
$│X-a│$ will be the distance between $X$ and $a$.

For example, we will solve the following inequality:
$│X-2│<6$
What does this inequality mean?

Basically, we are looking for all the $X$ that exist where the distance between $X$ and $2$ is less than $6$.

In the first step:
We will draw the number line and mark the point $2$, from which we are interested in the distance to $X$.
We will mark it in blue.

In the second step:
We will find the points whose distance to $2$ is exactly $6$. That is $8$ and $-4$.
We will mark them in green.

In the third stage:
We ask ourselves, what are we looking for?
In this example, we are looking for all the points whose distance to $2$ is less than $6$. That is, all the $X$ that are between $-4$ and $8$.

And we have arrived at the answer. The solution is  $-4<X<8$

An example of inequality with the sign greater than:
$│X-2│>6$
What does this inequality mean?
Basically, we are looking for all the $X$ that exist, where the distance between $X$ and $2$ is greater than $6$.
The first and second stages are exactly the same as in the previous inequality.
We draw a number line. $Y$ and we will mark the relevant points.
In the third stage:
We will ask ourselves, what are we looking for?
In this example, we are looking for all the points whose distance to $2$ is greater than $6$.
That is, all the $X$ that are greater than $8$ or the $X$ that are less than $-4$.
We will present the solution in this way:

Algebraic solution:
When we are given an inequality with absolute value, we do not know if the expression as an absolute value is positive or negative, and therefore we will have to divide it into two cases.
The first case: the expression inside the absolute value is greater than $0$.
And the second case: the expression inside the absolute value is less than $0$.

Let's see this with an example:

After having found for each section its own field, the solution to the inequality will be:
$4≤X<10$    or    $-2<X<4$

That is, all the $X$ that are in the field of the first section or in the field of the second section.

We will draw a number line to find these $X$ and accept that the solution is:
$-2<X<10$

Regardless of the method you choose to solve the inequality with the absolute value presented to you, you will get the same result.
As you have seen, the geometric form is much shorter than the algebraic form and leaves less room for error.
We recommend that you use the geometric form if it is a relatively simple inequality and there is no unequivocal requirement to solve the inequality algebraically.
Of course, now all you have to do is practice both methods so that the inequality with absolute value cannot surprise you.
Prepare for any scenario and practice all the steps over and over again to find the correct fields.
Good luck!

If you are interested in this article, you might also be interested in the following articles:

Positive numbers, negative numbers, and zero

Inequality (with absolute value)

The real line

Opposite numbers

Elimination of parentheses in real numbers

Addition and subtraction of real numbers

Multiplication and division of real numbers

In the blog of Tutorela you will find a variety of articles about mathematics.