Solve equations of the form |ax + b| = c by splitting into two cases — with step-by-step work and number line visualization.
|expression| = c
The absolute value of a quantity equals its distance from zero on the number line. When you set |expression| equal to a constant c:
If c > 0: Two equations — expression = c OR expression = −c. Usually gives two solutions.
If c = 0: One equation — expression = 0. Exactly one solution (the expression must be zero).
If c < 0: No solution — absolute value is always ≥ 0, so it can never equal a negative number.
After solving, substitute each answer back into the original equation to verify it works. This is especially important for:
• Equations where the absolute value is not fully isolated
• More complex forms like |f(x)| = g(x) (where g(x) could be negative)
• Rational or radical equations combined with absolute value
If substituting gives a false statement, that answer is an extraneous solution and must be discarded.
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