# Inequalities

An **algebraic inequality** is an inequality in which two members are linked by one of these signs:

< | less than | 2x − 1 < 7 |

≤ | less than or equal to | 2x − 1 ≤ 7 |

> | greater than | 2x − 1 > 7 |

≥ | greater than or equal to | 2x − 1 ≥ 7 |

The solution of an inequality is the set of values which verifies the variable in the inequality.

The **solution of the inequality** can be expressed by:

1. **A graphical representation**.

2. **An interval**.

2x − 1 < 7

2x < 8 x < 4

(-∞, 4)

2x − 1 ≤ 7

2x ≤ 8 x ≤ 4

(-∞, 4]

2x − 1 > 7

2x > 8 x > 4

(4, ∞)

2x − 1 ≥ 7

2x ≥ 8 x ≥ 4

[4, ∞)

## Properties of Inequalities

If the two members of an inequality are **added or subtracted** by **the same number**, the resulting **inequality is equivalent** to that given.

3x + 4 < 5 3x + 4 − 4 < 5 − 4 3x < 1

If the two members of an inequality are **multiplied or divided** by **the same positive number**, the resultant **inequality is equivalent** to the given one.

2x < 6 2x : 2 < 6 : 2 ;x < 3

If the two members of an inequality are **multiplied or divided by the same negative number**, the resultant inequality changes sense and it is **equivalent** to the given one.

−x < 5 ;(−x) · (−1) > 5 · (−1) x > −5