Intuition for dimensions I got confused on the web and on my previous answers.
Sometimes a line got 1 dimension because you just need one point to define every part of it, sometimes it's in 2 dimensions if you consider it in a plan and sometimes it's in 3 dimensions if you plot it considering x, y and z. So how many dimensions has a line for example ?
Also, I found out that an inequality changes the number of dimensions, for example a line x+y=1 is in 2 dimensions if I plot it considering x and y but as soon as I write x+y>1 it's not a line anymore but a plan, so the number of dimensions increased by 1 so it is now a plan with 3 dimensions ? I think that is not correct but I do not know how..
Can you give me some intuitions or method to understand when I got how many dimensions considering an equation.
NB:

 A: The dimension of an object should be thought of as the answer to the following question:

If I were on the object, how many independent "axes of motion" could I move in?

For example, if you were an ant crawling along a line, there is only one axis of motion you can move in:  forward/backward.  This is what we mean when we say that a line is 1 dimensional.  It doesn't matter whether the line is considered by itself, or is drawn in a plane, or is part of a 3D environment:  on the line there is only one axis of motion, so the line is 1 dimensional.
If you were an ant crawling on a plane, or on the surface of a sphere, there are two axes of motion:  (1) forward/backward and (2) left/right.  These directions are perpendicular to each other, and of course you can move in a mixture of them as well.  But the fact that there are two independent axes of motion makes a plane (or the surface of a sphere) a 2-dimensonal object.
If, on the other hand, you were a bee buzzing around in mid-air, or in the interior of a spherical balloon, you would have three independent axes of motion:  (1) forward/backward, (2) left/right, and (3) up/down.  So those are 3-dimensional environments.
The number of dimensions an object has is an intrinsic propery of the object, and does not depend on where the object is.  We can put a 2-dimensional object in a 3-dimensional space (imagine a piece of paper blowing in the breeze).  We can also bound a 3-dimensional space by a 2-dimensional surface (think of the water contained inside a water balloon).  Similarly, a line (or indeed any curve, like the circumference of a circle) is 1-dimensional, regardless of whether it is contained by a 2-dimensional space or a 3-dimensional space, and regardless of whether it bounds a 2-dimensional region.
Edited to add:  The OP has added some clarification to the original question, and I now think I better understand the issue, so let me take a shot at addressing it.
What the OP wants to know is, "How do we think about the geometry of an equation?"  For example, how is it that the equation $x = 1$ defines a point on $\mathbb R$, a line in the plane $\mathbb R^2$, and a plane in $\mathbb R^3$?
The answer, I think, is that when we write an equation like
$$x = 1$$
we are abbreviating a slightly more complicated question.  The actual question is

Find the set of all solutions, in some "universe" $S$, to the equation $x = 1$.

We usually don't specify what the universe is, because context generally makes it clear.  However, when we move from one universe to another, suddenly the question "What kind of solutions are we looking for?" becomes very important.
Before I return to the question of dimension, let me illustrate the general point.  Consider the following equations:

*

*$3x - 5 = 0$.  If we are searching for solutions in the set of natural numbers, $\mathbb N$, then this equation has no solutions.  On the other hand if we are searching in the set of rational numbers, $\mathbb Q$, then this equation has one solution.

*$x^2 - 5 = 0$.  If we are working over $\mathbb Q$, this equation has no solutions.  On the other hand if we are working over $\mathbb R$, the set of real numbers, then it has two solutions.

*$x^2 + 5 = 0$.  Over $\mathbb R$, this has no solutions; over $\mathbb C$, the complex numbers, it has two solutions.

So you can see that the solution set for a particular equation depends not only on the equation itself, but also on the "types" of solutions we are willing to admit.
Now let's return to the question in the OP.  Consider the following three similar-looking questions:

*

*Find all $x \in \mathbb R$ such that $x = 5$.

*Find all $(x, y) \in \mathbb R^2$ such that $x = 5$.

*Find all $(x, y, z) \in \mathbb R^3$ such that $x = 5$.

The solution to the first problem is a point; the solution to the second problem is a line; and the solution to the third problem is a plane.  The equation is the same in all three cases, but the context is different.
To connect this to my earlier answer, let's consider the third case:  Suppose you are sitting at a point $(x, y, z) \in \mathbb R^3$ in the solution set of $x = 5$.  How much freedom of movement do you have, if you want to stay in that solution set?  You can't change your $x$-coordinate, as that has to stay equal to $5$.  But the equation says nothing about the values of $y$ or $z$; they are unconstrained by the equation.  So you can freely move in the $y$ direction (left/right motion) or in the $z$ direction (up/down motion).  Two axes of freedom, hence you are sitting in a 2-dimensional object.
On the other hand, if you are sitting at a point $(x, y) \in \mathbb R^2$ in the solution set of $x = 5$, the only freedom of motion you have is in the $y$ direction.  One axis of freedom, therefore a 1-dimensional object.
Very loosely speaking, an equation in $n$ variables imposes a constraint on the variables, which means you can choose $n-1$ of the variables to have any values you want, but then the final variable's value is determined by the constraint. The result is then (typically) a surface of $n-1$ dimensions.  If you have a system of two equations in $n$ variables, each equation imposes a constraint, so you have (again, typically) only $n-2$ free variables left, resulting in a surface of dimension $n-2$.
A: You need to think about how many independent things you need to specify in order to fully specify a point on your line (or other mathematical object).
A line itself (at least, at the high school level) is a 1D object with equation of the form $y = mx + b$.  Once we specify $x$, we automatically specify a $y$ value.
If we now choose to plot that line in the plane, then it is a 1D object that lives in 2D space (or 3D, or whatever).  It is still a 1D object, though, and we can see that by parameterizing it.  A line in the plane can be thought of as two equations:
$$x(t) = t$$
$$y(t) = mt + b$$
Now, although there are technically two functions, they both rely on a single variable, and as such the object is 1D.  The two functions allow us to essentially draw it in our 2D space. As such, the "set of points that lie on the line" is a set of 2D mathematical objects known as points, but the line itself is a 1D object.   You are free to think of it in either 1D (a single equation that produces y values) or 2D (a set of equations that produces points) whenever it is convenient.
When we include an inequality, let's think about what happens:  $x + y > 3$.  Here, we may consider this to be a region in the plane or a higher dimensional object, depending on what we want.
As a planar segment: $x+y > 3 \implies y > 3 - x$, therefore this specifies a region in the plane that contains, for all $x$ values, vertical line segments who start at $y = x-3$ and go to infinity.
As a segment of a higher dimensional space: if it is convenient, we might claim that $z = x+y$ is a 3D object, in this case a plane living in 3D*. Then $x+y>3$ specifies all points on that plane whose $z$ values are greater than 3.
*and if it is instead convenient, we may consider it to be a 2D object!
The actual answer is: it depends on what you need it to be.  If you need the line to be in 1D, then so be it.  If you need to plot that line in 2D or something else, then you need to think of it as an object living in that space, appropriately.
