Special Cases: Impossible or Infinite Solutions
Recognize when absolute value inequalities yield no solution or all real numbers. Identify impossible and infinite solution cases in Grade 9 algebra.
Key Concepts
Property If an inequality simplifies to $|K| \le (\text{negative number})$, there is no solution ($\{\}$ or $\emptyset$). If it simplifies to $|K| \ge (\text{negative number})$, the solution is all real numbers ($\mathbb{R}$). Explanation This is a logic check! Absolute value represents a distance, and distance cannot be negative. Asking for a distance to be less than or equal to 2, like in $|x| \le 2$, is impossible. On the flip side, asking for a distance to be greater than 5 is always true for any real number. Examples $|x| + 6 \le 4$ simplifies to $|x| \le 2$. Since absolute value cannot be negative, there is no solution, $\emptyset$. $|x| + 6 1$ simplifies to $|x| 5$. Since absolute value is always non negative, this is true for all real numbers, $\mathbb{R}$.
Common Questions
When does an absolute value inequality have no solution?
When it simplifies to absolute value of expression less than or equal to a negative number. Since absolute value is always non-negative, no real number satisfies it.
When does an absolute value inequality have all real numbers as solutions?
When it simplifies to absolute value of expression greater than or equal to a negative number. Since absolute value is always at least 0, every real number works.
How do you identify these special cases?
Isolate the absolute value expression first. If the right side is negative with less-than-or-equal, the answer is no solution. If negative with greater-than-or-equal, all reals.