Question

Describe the set of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. \[{x^2} + {y^2} + {z^2} \le 1\]

Answer 1

Hint:
Here, we have to describe the set of points in the space satisfying the given inequality. Here we will analyze the given inequality and find its general equation. Then we will compare it to the general equation and find the radius and thus the sets of points. An equation is a relative comparison between any two mathematical expressions that are equal. Inequality is a relative comparison between two mathematical expressions which may be greater than or lesser than or greater than equal to or lesser than equal to.

Complete Step by Step Solution:
We are given an inequality \[{x^2} + {y^2} + {z^2} \le 1\] .
We know that the equation of the sphere having centre at the origin \[\left( {0,0,0} \right)\] is given by \[{x^2} + {y^2} + {z^2} = {r^2}\] where \[\left( {x,y,z} \right)\] are the coordinates of the sphere and \[r\] is the radius of the sphere.
So, we have \[{x^2} + {y^2} + {z^2} = 1\]
The given equation is in the form of the equation of the sphere with radius as 1 unit and centre at the origin \[\left( {0,0,0} \right)\].
Thus, for an inequality \[{x^2} + {y^2} + {z^2} \le 1\] , the points lie within the sphere of radius as 1 unit.

Therefore, the inequality \[{x^2} + {y^2} + {z^2} \le 1\] represents the set of points which lie within the sphere of radius as 1 unit.

Note:
We should remember that we are finding the set of points that satisfy the given inequality and we are not solving for the variables. Whenever we are in need to find the set of points, we should also know the general equation of the various geometrical shapes. But when we want to solve an inequality it is essential to find all the variables that satisfy the given inequality and make it true.