Quartic Inequality Solver

Solve a quartic inequality ax to the 4th plus bx cubed plus cx squared plus dx plus e. Get the real roots and the solution intervals on a number line.

Input

Enter the coefficients of the quartic inequality and a relation to find the range of x that solves it.

a x to the 4th + b x³ + c x² + d x + e ＞ 0

Coefficient a (x to the 4th)

Coefficient b (x³)

Coefficient c (x²)

Coefficient d (x)

Constant e

Relation

Result

Solution range

x is less than -2 or greater than -1 and less than 1 or x is greater than 2

Real roots

-2 , -1 , 1 , 2

Degree

Number line

Solution intervals

Interval 1	( negative infinity , -2 )
Interval 2	( -1 , 1 )
Interval 3	( 2 , positive infinity )

How it works

Enter the five coefficients and a relation to find every x for which ax to the fourth plus bx cubed plus cx squared plus dx plus e satisfies the chosen relation with zero.
The tool first finds the real roots of the polynomial numerically, splits the number line at the distinct roots, and checks the sign at a sample point inside each interval to decide which intervals belong to the solution.
With at least or at most, the roots themselves are part of the solution and the interval ends are closed. With greater or less, the roots are excluded and the ends are open.
If the leading coefficient is zero, the expression is treated as a lower degree polynomial and solved the same way. If every coefficient is zero so no variable remains, the inequality cannot be solved.
Because the roots are computed numerically, very close roots or the neighborhood of a repeated root may show a tiny rounding difference.

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