Special Cases: Zero and Negatives
Understand special cases in absolute value equations for Grade 9 math: when |A|=0 has exactly one solution, and when |A| equals a negative number yields no solution.
Key Concepts
Property If $|A|=0$, the only solution is $A=0$. If $|A|$ equals a negative number, there is no solution, which is written as $\emptyset$. Explanation Watch out for these tricky special cases! If an absolute value equals 0, there is only one possible answer because only 0 is 0 units away from itself. If an absolute value equals a negative number, it's a trap! Distance can never be negative, so there is no solution at all. Don't waste your time solving it. Examples The equation $|x 6|=0$ has only one solution because $x 6$ must be 0, so $x=6$. The equation $|x+4| = 8$ has no solution, or $\emptyset$, because an absolute value cannot equal a negative number. If you encounter $|3y+1| = 5$, you can immediately state that the solution set is empty without doing any algebra.
Common Questions
What are the special cases in absolute value equations?
If |A| = 0, the only solution is A = 0, since only zero is zero units from itself. If |A| equals a negative number, there is no solution (written as the empty set), because absolute values represent distance and cannot be negative.
How do you solve |x - 6| = 0 versus |x + 4| = -8?
|x - 6| = 0 has exactly one solution: x = 6, because x - 6 must equal zero. In contrast, |x + 4| = -8 has no solution at all, since an absolute value can never equal a negative number — you can state the solution set is empty without any further algebra.
Why is it important to recognize these special cases in Grade 9 algebra?
Recognizing special cases saves time and prevents errors. When you see an absolute value set equal to a negative number, you can immediately identify no solution exists. These special cases appear frequently in Chapter 8 Advanced Factoring and Functions and on assessments.