Linear Inequalities — Formula Sheet

Equal numbers may be added to or subtracted from both sides of an inequality without affecting the sign.

Both sides of an inequality can be multiplied or divided by the same positive number.

When both sides are multiplied or divided by a negative number, the sign of inequality is reversed.

x < 2

x > -2

x ≥ 8

x < 3

-1 ≤ x < 2

5x - 3 < 3x + 1 => x < 2

4x + 3 < 6x + 7 => x > -2

(5 - 2x)/3 ≤ x/6 - 5 => x ≥ 8

7x + 3 < 5x + 9 => x < 3

Examples of literal inequalities

x < 5, y > 2, x ≥ 3, y ≤ 4.

40x + 20y ≤ 120 is a slack inequality.

40x + 20y = 120 is an equation.

For example, ax + b < 0 is strict, while ax + b ≤ 0 is slack.

Expressions of the form ax + b < 0, ax + b > 0, ax + b ≤ 0, or ax + b ≥ 0 with a ≠ 0 are linear inequalities in one variable.

Expressions of the form ax + by < c, ax + by > c, ax + by ≤ c, or ax + by ≥ c with a ≠ 0 and b ≠ 0 are linear inequalities in two variables.