Inequalities with Absolute Value

• We will draw a number line, and mark the point from which we are interested in the distance to $X$.
• Then we find the points whose distance from the relevant point is exactly the distance mentioned in the condition
• Now, let's pay attention to the condition: $>$ or $<$ And we will find the results.

To solve the absolute value inequality using the algebraic method, we will divide it into two cases.
In the first case, we will consider that the expression inside the absolute value is positive and greater than zero.
In the second case, we will consider that the expression inside the absolute value is negative, less than zero.

Then, we will return to the original inequality and divide it into two cases as well.
In the first case, we can simply remove the absolute value (since it is a positive expression) and solve the inequality in the usual way.
In the second case, we will have to remove the minus sign from outside the expression to get rid of the absolute value, open parentheses, and then solve the inequality.

We will find for each case the values that also meet the conditions - the common domain for each case.
To do this, we will draw a number line and mark the fields accordingly.

Now we find all the values of $X$ that have the common domains, all the values that are in this domain satisfy the inequality.

Example to solve inequalities:
$3x+2<4x-13$
We will move the section:
$-x<-15$
Divide by 1 both sections and remember to convert the inequality sign as follows:
$x>15$
We have reached the solution of the inequality. We find all the values of $X$ that satisfy the inequality.
The meaning of the result is that when we set any $X$ greater than $15$, the inequality will exist.

Basic rule:
$│x-a│ =$ the distance between $X$ and $a$.

To solve the absolute value inequality using the geometric method, we will need to find all the values for which the distance between them and a satisfies the condition.
How do we do that?
Let's see the solution through an example:
$│x-3│<7$
We are asked, what are the values of $X$, whose distance between them is $3$, less than $7$.
We draw a number line, and mark the point of interest for the distance to $X$. In this example $-3$.
Then we will find the points whose distance from the relevant point (in this example $-3$) is exactly the distance mentioned in the condition (in this example the distance is $7$) and mark it in a different color.

Now, let's pay attention to the condition: $>$ OR $<$ . In this example, we need to find the points whose distance from $3$ is less than $7$.
That is, all the values between $10$ and $-4$.
We will mark them on the figure and the result will be: $-4<X<10$
If the inequality were the same only with $7>$, the result is
$x>10$ or $x<-4$
Since these are the values whose distance from $3$ is greater than $7$.

We will take the same inequality that we solved geometrically and now solve it algebraically:
To solve the absolute value inequality using the algebraic method, we will divide it into two cases.
$│x-3│<7$
In the first case, we will consider that the expression inside the absolute value is positive and greater than zero.
In the second case, we will consider that the expression inside the absolute value is negative, less than zero.
We will solve the inequalities we have obtained.

First case:
$X-3\ \geq 0$
That is
$X\ \geq3$

Second case:
$X-3<0$
That is
$X<3$

Then, we will return to the original inequality and divide it into two cases as well.
In the first case, we can simply remove the absolute value (since it is a positive expression) and solve the inequality in the usual way.
In the second case, we will have to add a minus outside the expression to get rid of the absolute value, open parentheses, and then solve the inequality.
We will mark the final solutions with a different color.

Continuing with the first case:
$X-3<7$
$X<10$

Continuing with the second case:
$-(X-3)<7$
$-X+3<7$
$-X<4$
$X<-4$

Pay attention! We have not finished the solution yet.
To continue, we will find in each case the values that also meet the conditions: the common domain for each case.
To do this, we will draw a number line and mark the domains accordingly when a solid point includes the number and an open point does not include the number.
After marking the domains, we find the common domain for both domains and mark it with a different color.

Common domain:
$-4<X<3$

Common domain:
$3\ \leq X>10$

And again... we are not done yet.
Now we will find all the values of $X$ that contain the common domain (yellow) of the first case or the common domain (yellow) of the second case.

We will draw the number line again and it can be easily seen:

We can see that this solution to the inequality is $-4<X<10$
All values within this range maintain the inequality.
Note that we obtained the same result both algebraically and geometrically.
Of course, no matter which method you choose, if you follow the steps and do not make mistakes, you will arrive at the correct answer one way or another.

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

Positive numbers, negative numbers, and zero

Absolute value and 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 Tutorela blog, you will find a variety of articles about mathematics.