Solve absolute value equations of the form a|x + b| = c and find all solutions.
Real-number coefficients only (decimals or scientific notation); complex numbers are not supported here.
An absolute value equation is an equation that contains an absolute value expression. The absolute value of a number is its distance from zero on the number line, which is always non-negative. An equation like |x| = 5 means "x is 5 units away from zero," which gives two solutions: x = 5 or x = -5.
More generally, equations of the form a|x + b| = c can be solved by recognizing that the expression inside the absolute value can be either positive or negative. This leads to two separate linear equations that must be solved independently.
The solver automatically divides by coefficient a so the absolute value term becomes isolated even when a ≠ 1. When you take these same steps to inequalities, dividing by a negative a flips the inequality direction, though equalities (like the ones solved here) stay unchanged.
Solve: 2|x - 3| = 10
Yes. If the equation requires the absolute value to equal a negative number (e.g., |x| = -3), there is no solution because absolute values are always non-negative.
Yes. This happens when the right side equals zero (e.g., |x| = 0 gives only x = 0) or when both cases yield the same solution.
Not always. You may get two distinct solutions, one solution, or no solution depending on the equation.
You'll need to consider multiple cases based on the signs of the expressions inside each absolute value. This can lead to up to four separate equations to check.
Substitute each solution back into the original equation. If both sides are equal, the solution is correct. Always check your work!
The equation |x| = 5 has exactly two solutions: x = 5 or x = -5. The inequality |x| < 5 has infinite solutions: all x between -5 and 5.
Extraneous solutions are values that arise when solving an equation but don't satisfy the original equation. Always check your solutions by substituting back into the original equation.
Yes! They model situations involving distance, tolerance, error bounds, and any scenario where you need to find values at a fixed distance from a target. Common in engineering and physics.
No, this calculator only accepts real-number coefficients (decimals or scientific notation). Complex tasks need a symbolic or complex-number solver.
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