Addition Property of Inequality

Property For any real numbers $a$, $b$, and $c$: If $a < b$, then $a + c < b + c$. If $a b$, then $a + c b + c$. If $a \leq b$, then $a + c \leq b + c$. If $a \geq b$, then $a + c \geq b + c$. Explanation Think of an inequality as a wobbly seesaw. As long as you add the same weight (or number) to both sides, it stays tilted in the same direction! This property is your secret weapon for solving inequalities. It lets you isolate the variable by adding the same value to both sides without messing up the inequality’s truth. Examples To solve $x 12 < 8$, we add 12 to both sides: $x 12 + 12 < 8 + 12$, which simplifies to $x < 4$. To solve $y 5 \geq 2$, we use the same magic: $y 5 + 5 \geq 2 + 5$, which simplifies to $y \geq 7$. For a fractional challenge like $z \frac{1}{2} 5$, just add $\frac{1}{2}$ to each side: $z \frac{1}{2} + \frac{1}{2} 5 + \frac{1}{2}$, resulting in $z 5\frac{1}{2}$.