Equations with absolute values

To solve equations with absolute values, we consider the definition: $|A| = B$ means $A = B$ or $A = -B$, provided $B \geq 0$. For more complex equations, like $|x - 2| = 5$, we set $x - 2 = 5$ or $x - 2 = -5$, so $x = 7$ or $x = -3$. For equations like $|2x + 1| = |x - 4|$, we consider both cases: $2x + 1 = x - 4$ and $2x + 1 = -(x - 4)$.

Important properties

• If $|A| = B$ and $B < 0$, there is no solution.

• If $|A| = B$ and $B \geq 0$, solutions are $A = B$ or $A = -B$.

• For $|A| = |B|$, solutions are $A = B$ or $A = -B$.

• Always check solutions in the original equation, especially if you square both sides.