About the Compound Inequality Solver
The Compound Inequality Solver finds the range of x that satisfies a three-part inequality of the form a < bx + c < d, returning the solution as a simple lower-than-x-than-upper range. It's built for algebra students working through compound inequality problems, where solving both bounds simultaneously by hand is easy to get wrong, especially when the coefficient on x is negative.
How It Works
You enter the lower bound (a), the coefficient of x (b), the constant term (c), and the upper bound (d) from an inequality written as a < bx + c < d. The calculator isolates x by subtracting c from every part of the inequality and then dividing every part by b, exactly as you would solving it by hand. If b is negative, dividing by it flips both inequality signs, so the calculator swaps the two resulting bounds to keep the range correctly ordered from smaller to larger.
Formula & Methodology
Solving by hand follows the same two steps the calculator performs: subtract c from all three parts of the inequality to get a - c < bx < d - c, then divide all three parts by b, which requires flipping both inequality symbols if b is negative. The two resulting values are the interval's endpoints, and writing the smaller one first with the correct inequality direction gives the final solution range.
Examples
Positive Coefficient
For 1 < 2x + 3 < 11, subtracting 3 from each part gives -2 < 2x < 8, and dividing by 2 gives -1 < x < 4, which is exactly what the calculator returns for its default values.
Negative Coefficient
For -5 < -3x + 4 < 7, subtracting 4 gives -9 < -3x < 3, and dividing by -3 flips both inequality signs and swaps the bounds, giving -1 < x < 3.
Advantages
- Automatically flips and reorders the bounds when the coefficient of x is negative, which is the step most commonly done incorrectly by hand.
- Solves the entire three-part inequality in one pass instead of requiring you to split it into two separate inequalities first.
- Returns a cleanly ordered lower-to-upper range regardless of the sign of any input value.
Common Mistakes
- Forgetting to flip both inequality signs when dividing by a negative coefficient, which produces a range with the bounds in the wrong order or the wrong direction.
- Subtracting or dividing only part of the inequality, for example adjusting bx + c but not also adjusting both a and d by the same operation.
- Treating a and d as if they can be entered in either order, when the calculator takes a as the lower bound and d as the upper bound of the original inequality as written, before any solving happens.
Edge Cases to Watch For
- If b equals 0, there's no x term left to solve for, so the calculator returns an error instead of a range, since bx + c would just reduce to the constant c regardless of x.
- When b is negative, both computed bounds are swapped before display, since dividing an inequality by a negative number reverses the direction of both inequality signs at once.
- The solver assumes a strict inequality using less-than rather than less-than-or-equal; it reports the same numeric bounds either way, since the underlying algebra for isolating x is identical for strict and non-strict inequalities.
- The calculator takes whatever values are entered for a and d as the original lower and upper bounds before solving, so double-checking that a problem was transcribed with the smaller bound on the left avoids a mismatched setup.
Common Use Cases
- Algebra students solving three-part compound inequality problems as homework or exam practice.
- Teachers or tutors generating quick answer keys to check a batch of compound inequality problems.
- Anyone verifying a hand-solved inequality range, particularly to confirm the sign-flip step was applied correctly.