Properties of Inequalities

The following are the properties of inequalities:

Properties of inequalities
Addition propertyFor the real numbers x, y y z,• If x<y, then x+z<y+z.
Subtraction propertyFor the real numbers x, y y z,• If x<y, then x-z<y-z.
Multiplication propertyFor 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 propertyFor the real numbers y y,• If x<y, then y≮ x.• If x>y, then y≯ x.
Transitive propertyFor 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.
how to solve linear inequalities

Relevant for

Learning about the properties of inequalities.

See definitions

how to solve linear inequalities

Relevant for

Learning about the properties of inequalities.

See definitions

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:

properties of addition and subtraction of inequalities

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


  • 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.

properties of multiplication and division of inequalities

$latex x>y$ 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:


  • 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:


  • 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.

transitive property of inequalities

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 $latex x>z$.


  • 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, $latex y>x$


  • 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:

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