The following are the properties of inequalities:
| Properties of inequalities |
| Addition property | For the real numbers x, y y z,• If x<y, then x+z<y+z. |
| Subtraction property | For the real numbers x, y y z,• If x<y, then x-z<y-z. |
| Multiplication property | For the real numbers x, y y z,• If x<y, then:• xz<yz, only if z>0• xz>yz, only if z<0• xz=yz, only if z=0 |
| Antisymmetric property | For the real numbers x y y,• If x<y, then y≮ x.• If x>y, then y≯ x. |
| Transitive property | For the real numbers x, y y z,• If x<y y y<z, then, x<z.• If x>y y y>z, then, x>z. |
Properties of addition and subtraction
When we add z to both sides of the inequality, we are simply moving the whole inequality, so the inequality remains the same:
If $latex x>y$, then, $latex x+z>y+z$
Similarly, we have the following:
- If $latex x>y$, then $latex x-z>y-z$
- If $latex x<y$, then $latex x+z<y+z$
- If $latex x<y$, then $latex x-z<y-z$
This means that adding or subtracting the same value from both x and y will not change the inequality
EXAMPLE
- Carl has less money than David.
If Carl and David receive 5 dollars each, Carl still has less money than Matías. The relationship has not changed.
Properties of multiplication and division
When we multiply both x and y by a positive number, the inequality remains the same.
However, when we multiply both x and y by a negative number, the inequality flips.
$latex x>y$ becomes $latex x<y$ when multiplying by -2
But the inequality remains the same when multiplying by 2
These are the general rules:
- If $latex x<y$ and z is positive, then $latex xz<yz$
- If $latex x<y$ and z is negative, then $latex xz>yz$ (the sign changes)
The following is an example of multiplication by a positive number:
EXAMPLE
- Diana got a grade of 4 which is less than the grade of 5 that Andres got.
$latex x<y$
If both Diana and Andrés manage to double their grade (multiply by 2), Carolina’s grade will continue to be lower than Andrés’s grade.
$latex 2x<2y$
Now let’s see what happens when multiplying by a negative:
EXAMPLE
- If the grading turn negative (multiply by -1), then Diana loses 4 points and Andres loses 5 points.
This means that Diana now gets a higher grade than Andres.
$latex -x>-y$
Transitive property
When we relate inequalities in order, we can skip the inequality in the middle.
If we have $latex x<y$ and $latex y<z$, then $latex x<z$.
Similarly, if we have $latex x>y$ and $latex y>z$, then $latex x>z$.
EXAMPLE
- If Jhon is older than Richard and,
- If Richard is older than Sergey,
Therefore, Jhon must be older than Sergey.
Antisymmetric property
The values x and y cannot be swapped if we keep the same inequality sign.
- If we have $latex x>y$, this is different than $latex y>x$. Then, we have $latex y\ngtr x$
- If we have $latex x<y$, this is different than $latex y<x$. Then, we have $latex y\nless x$
If we swap the x and y values, we must make sure to change the inequality sign:
- If $latex x>y$, then, $latex y<x$
- If $latex x<y$, then, $latex y>x$
EXAMPLE
- If Jhon is older than Richard, then Richard is younger than Jhon.
See also
Interested in learning more about inequalities? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
