The ``forget the minus sign" definition of the absolute value is useless for our purposes. Instead, we will mostly use the geometric definition of the absolute value:

**The absolute value of a number measures its distance to the origin
on the real number line.**

Since 5 is at 5 units distance from the origin 0, the absolute value of 5 is 5, |5|=5

Since -5 is also at a distance of 5 units from the origin, the absolute value of -5 is 5, |-5|=5:

We are ready for our first inequality. Find the set of solutions for

|*x*|<5.

Translate into English: we are looking for those real numbers *x* whose distance from the origin is less than 5 units.

Obviously we are talking about the interval (-5,5):

What about the solutions to ?

In English: which numbers, *x*, are at least 2 units away from the
origin? On the left side, real numbers less than or equal to -2
qualify, on the right all real numbers greater than or equal to 2:

We can write this interval notation as

What is the geometric meaning of |*x*-*y*|?

|*x*-*y*|** is the distance between x and y on the real number line.**

Consider the example |(-4)-3|. The distance on the real number line
between the points -4 and +3 is 7, thus

|(-4)-3|=7.

Let's find the solutions to the inequality:

In English: Which real numbers are not more than 1 unit apart from 2?

We're talking about the numbers in the interval [1,3].

What about the example

Let's rewrite this as

which we can translate into the quest for those numbers

The set of solutions is

With a little bit of tweaking, our method can also handle inequalities
such as

|2*x*-5|<2.

We first divide both sides by 2. Note that absolute values interact nicely with multiplication and division:

as long as

Thus we obtain

after simplification, we get the inequality

asking the question, which numbers are less than 1 unit apart from

So the original inequality has as its set of solutions the interval .

Consider the example

Let's divide by 3:

which is the same as

Which numbers have distance at least from ? The set of solutions is given by

Our method fails for more contrived examples.

Let us consider the inequality

|*x*-3|<2*x*-4

It's back to basic algebra with a twist.

The standard definition for the absolute value function is given by:

Thus we can get rid of the sign in our inequality if we know whether the expression inside,

We will do exactly that!

Let's first consider only those values of *x* for which :

**Case 1**:

In this case we know that |*x*-3|=*x*-3, so our inequality becomes

Solving the inequality, we obtain

We have found some solutions to our inequality:

x is a solution if
and *x*>1 at the same time! We're talking
about numbers .

What if *x*-3<0?

**Case 2**: *x*<3

This time *x*-3<0, so
|*x*-3|=-(*x*-3)=3-*x*,
so our inequality reads as

3-*x*<2*x*-4.

Applying the standard techniques, this can be simplified to

Our inequality has some more solutions:

Under our case assumption *x*<3, solutions are those real numbers
which satisfy
.

We're talking about numbers in the interval

Combining the solutions we found for both cases, we conclude that the set of solutions for the inequality

|*x*-3|<2*x*-4

are the numbers in the interval

|*x*-3|>5.

|2*x*-5|>*x*+1.

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