No Solution Case for Absolute Value Equations
Absolute value equations of the form |ax + b| = c have no solution when c < 0, because absolute value always produces a non-negative result. In Grade 11 enVision Algebra 1 (Chapter 1: Solving Equations and Inequalities), students learn to check the right-hand side of any absolute value equation before solving: if c is negative, they can immediately conclude no solution exists without further calculation. This check prevents wasted work and reinforces the geometric interpretation of absolute value as distance from zero, which can never be negative.
Key Concepts
If $|ax + b| = c$ where $c < 0$, then the equation has no solution because absolute value expressions are always non negative: $|expression| \geq 0$.
Common Questions
Why does |ax + b| = c have no solution when c < 0?
Absolute value represents distance from zero, which is always zero or positive. Setting it equal to a negative number is a contradiction with no real solution.
How do you identify the no-solution case for absolute value equations?
Check the value on the right side of the equation. If it is negative (c < 0), there is no solution immediately — no further solving is needed.
Does |ax + b| = 0 have a solution?
Yes. When c = 0, there is exactly one solution: set ax + b = 0 and solve for x.
What is the first step when solving any absolute value equation?
Isolate the absolute value expression on one side, then check whether the other side is negative (no solution), zero (one solution), or positive (two solutions).
Can you solve |x − 3| = −5?
No. Since −5 < 0 and absolute value cannot be negative, this equation has no solution.
How does the number of solutions relate to the value of c in |ax + b| = c?
If c < 0: no solution. If c = 0: one solution. If c > 0: two solutions.