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:

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