Solutions of Inequalities

An equation pins down a set of points; an inequality carves out a region. This tutorial is a hands-on tour of how Mathilda solves inequalities — from a single quadratic in one variable through rational functions, absolute values and piecewise functions, and on to two- and three-variable regions of the plane and space. The one tool at the centre of it is Reduce, introduced in the solutions of equations tutorial; here it does the thing Solve cannot, because the answer is almost never a finite list of values.

Every transcript below was produced by the actual Mathilda binary. Type the In[...] lines yourself (without the prompt) and you will see the same Out[...]. As with equations, inequalities use the ordinary comparison operators <, <=, >, >= (and != for "not equal"); a chained form such as -5 < 3 x + 7 <= 22 means both parts at once. The answer comes back as a logical combination — && ("and"), || ("or") — of simpler inequalities describing the whole solution set.

A word on domains

Ordering (<, >, …) is undefined over the complex numbers, so a statement that contains an inequality and is given no explicit domain is solved over the reals — matching Mathematica. That means the third argument Reals is optional whenever the statement already contains an inequality; the examples below often include it for clarity, but Reduce[x^2 < 4, x] and Reduce[x^2 < 4, x, Reals] mean exactly the same thing.

Sign diagrams: one polynomial inequality

To solve a polynomial inequality, Reduce finds the real roots of the polynomial — the points where it can change sign — and reports the union of the intervals on which the inequality holds. This is the classic sign diagram you draw by hand, done exactly:

In[1]:= Reduce[x^2 > 1, x, Reals]
Out[1]= x < -1 || x > 1

In[2]:= Reduce[x^2 - 5 x + 6 < 0, x, Reals]
Out[2]= 2 < x < 3

In[1] splits the line at the roots ±1 and keeps the two outer intervals where x² − 1 > 0. In[2] factors as (x − 2)(x − 3) < 0, which holds only between the roots. Higher degrees are no harder — the diagram just has more breakpoints:

In[1]:= Reduce[(x - 1) (x - 2) (x - 3) > 0, x, Reals]
Out[1]= 1 < x < 2 || x > 3

In[2]:= Reduce[x^4 - 5 x^2 + 4 < 0, x, Reals]
Out[2]= -2 < x < -1 || 1 < x < 2

The cubic alternates sign across its three roots, so the positive set is every other interval; the quartic (x² − 1)(x² − 4) is negative on the two bands between consecutive roots. When the breakpoints are irrational, Reduce keeps them exact rather than rounding — it orders and signs algebraic numbers with an exact real-algebraic oracle:

In[1]:= Reduce[x^2 < 2, x, Reals]
Out[1]= -Sqrt[2] < x < Sqrt[2]

A <= or >= closes the corresponding endpoint, and != punches out isolated points:

In[1]:= Reduce[x^2 <= 4, x, Reals]
Out[1]= -2 <= x <= 2

In[2]:= Reduce[x^2 != 1, x, Reals]
Out[2]= x != -1 && x != 1

Combining conditions

Because the result is a logical formula, you can hand Reduce any combination of inequalities joined with && and ||, and it intersects and unions the solution sets accordingly. A chained inequality is just two bounds at once:

In[1]:= Reduce[-5 < 3 x + 7 <= 22, x]
Out[1]= -4 < x <= 5

In[2]:= Reduce[x^2 > 1 && x < 3, x, Reals]
Out[2]= x < -1 || 1 < x < 3

In[3]:= Reduce[1 <= x^2 <= 4, x, Reals]
Out[3]= -2 <= x <= -1 || 1 <= x <= 2

Note that In[1] was given no domain: because it contains an ordering, it is solved over the reals automatically. In[3] — the set where x² lies between 1 and 4 — is a good illustration of why the answer is a formula and not a list: it is the union of two symmetric closed bands.

Rational functions and their poles

A rational inequality cannot be solved by "multiplying out" the denominator — that would flip the inequality wherever the denominator is negative, and would wrongly include the points where it vanishes. Reduce instead treats the roots of the denominator as poles: extra breakpoints where the expression is undefined and which are excluded from the solution set.

In[1]:= Reduce[1/x < 1, x, Reals]
Out[1]= x < 0 || x > 1

In[2]:= Reduce[1/x >= 0, x, Reals]
Out[2]= x > 0

In[3]:= Reduce[(x - 1)/(x - 2) > 0, x, Reals]
Out[3]= x < 1 || x > 2

In[1] is the tell-tale case: 1/x < 1 is satisfied by every negative x (where 1/x is itself negative) and by x > 1, but not by the interval 0 < x <= 1. Naïvely clearing the denominator would have given the wrong x > 1 alone. In In[2] the pole at x = 0 is excluded, so the answer is the open ray x > 0, not x >= 0 — 1/x is undefined at the origin.

Deciding a statement: True and False

Some inequalities hold for every value of the variable, or for none. There the "solution set" is the whole line or the empty set, and Reduce returns a bare True or False — a genuine proof, not a sampled guess:

In[1]:= Reduce[x^2 + 1 > 0, x, Reals]
Out[1]= True

In[2]:= Reduce[x^2 >= 0, x, Reals]
Out[2]= True

In[3]:= Reduce[x^2 < 0, x, Reals]
Out[3]= False

In[1] proves that x² + 1 is strictly positive for all real x; In[3] proves that a real square is never negative, so the constraint is contradictory.

Absolute values and piecewise functions

Reduce understands the real-valued constructs that make an expression piecewise-defined: Abs, Min/Max, Sign, Floor/Ceiling/Round, Mod, and the piecewise heads (Piecewise, UnitStep, Ramp, Clip, HeavisideTheta, …). It eliminates each one by case-splitting — an Abs on the sign of its argument, a Max on which argument is largest — solving each branch on its own sign diagram and unioning the results.

In[1]:= Reduce[Abs[x] < 1, x, Reals]
Out[1]= -1 < x < 1

In[2]:= Reduce[Abs[x] >= 2, x, Reals]
Out[2]= x <= -2 || x >= 2

In[3]:= Reduce[Abs[2 x - 1] <= 3, x, Reals]
Out[3]= -1 <= x <= 2

The method is not limited to a single absolute value. A sum of them — the kind of expression that defines a distance — is handled by splitting on every argument at once:

In[1]:= Reduce[Abs[x - 1] + Abs[x + 1] <= 4, x, Reals]
Out[1]= -2 <= x <= 2

In[2]:= Reduce[Min[x, 1 - x] > 1/4, x, Reals]
Out[2]= 1/4 < x < 3/4

In[3]:= Reduce[Sign[x - 1] < 0, x, Reals]
Out[3]= x < 1

Over the integers

Pass Integers as the domain and Reduce reports the integer solutions. When the inequalities bound the variable to a finite range, it enumerates them:

In[1]:= Reduce[x^2 < 10 && x > 0, x, Integers]
Out[1]= x == 1 || x == 2 || x == 3

The bounded real set 0 < x < Sqrt[10] contains exactly the integers 1, 2 and 3, and Reduce lists them explicitly.

Two variables: regions of the plane

With more than one variable the solution set is a region, and Reduce describes it as a triangular cascade: a range for the first variable, then bounds on the next that may depend on the first, and so on. A system of linear inequalities is handled by Fourier–Motzkin elimination:

In[1]:= Reduce[x + y < 1 && x > 0 && y > 0, {x, y}, Reals]
Out[1]= 0 < x < 1 && 0 < y < 1 - x

In[2]:= Reduce[2 x + y <= 4 && x >= 0 && y >= 0, {x, y}, Reals]
Out[2]= 0 <= x <= 2 && 0 <= y <= 4 - 2 x

Read In[1] as a sweep: x ranges over (0, 1), and for each such x, y runs from 0 up to 1 − x — exactly the open triangle with corners (0,0), (1,0), (0,1).

When the boundaries curve, the linear method no longer applies and Reduce switches to cylindrical algebraic decomposition (CAD): it partitions the plane into cells on which every defining polynomial keeps a constant sign, then keeps the cells that satisfy the statement and reads off their bounds.

In[1]:= Reduce[x y > 0, {x, y}, Reals]
Out[1]= x < 0 && y < 0 || x > 0 && y > 0

In[2]:= Reduce[y > x^2, {x, y}, Reals]
Out[2]= y > x^2

In[3]:= Reduce[y >= x^2 && y <= x + 2, {x, y}, Reals]
Out[3]= -1 <= x <= 2 && x^2 <= y <= 2 + x

In[1] is the pair of open quadrants where x and y share a sign. In[3] is the lens between the parabola y = x² and the line y = x + 2: the two curves meet at x = −1 and x = 2, and between them the region runs from the parabola up to the line.

A curved boundary brings a radical into the answer. The open unit disk, for instance, is x between −1 and 1 with y between ±√(1 − x²):

In[1]:= Reduce[x^2 + y^2 < 1, {x, y}, Reals]
Out[1]= -1 < x < 1 && -1/2 Sqrt[4 - 4 x^2] < y < 1/2 Sqrt[4 - 4 x^2]

The bound 1/2 Sqrt[4 - 4 x^2] is just Sqrt[1 - x^2] written in Mathilda's canonical surface form — a good reminder that Out[] shows the system's exact value, which is not always the tidiest way you would write it by hand. Closing the inequality to <= closes the region's boundary, and the x-range along with it:

In[1]:= Reduce[x^2 + y^2 <= 1, {x, y}, Reals]
Out[1]= -1 <= x <= 1 && -1/2 Sqrt[4 - 4 x^2] <= y <= 1/2 Sqrt[4 - 4 x^2]

The same machinery extends to three variables and beyond — an origin-centred solid ball x² + y² + z² < 1 comes back as a nested cascade of bounds, each inner variable limited by the outer ones.

Soundness: when Reduce declines

Reduce never returns a wrong or incomplete answer. When it cannot decide a sign or ordering exactly — most often because a symbolic parameter's sign is unknown — it leaves the input unevaluated rather than guess:

In[1]:= Reduce[a x + b > 0, x]
Out[1]= Reduce[b + a x > 0, x]

Solving a x + b > 0 for x means dividing by a — which flips the inequality when a < 0 and collapses it when a = 0. Without knowing the sign of a, there is no single correct direction, so Reduce declines. (Contrast this with the equation Reduce[a x == b, x] from the previous tutorial, which does return the full case tree, because an equation has no direction to flip.) A returned formula, then, is always guaranteed to describe the whole solution set exactly, and a returned False is a genuine proof that no solution exists.

Where to next

You can now solve polynomial, rational, absolute-value and piecewise inequalities in one variable, decide always-true and never-true statements, enumerate integer solutions, and describe multi-variable regions of the plane and space with Reduce.

• 7. Solutions of equations — the companion tutorial: Solve and Reduce on equations and systems, Root objects, Eliminate, and the transcendental solution families.
• 8. Calculus — where inequalities describe the domain of a function, the sign of a derivative, or the region of convergence you are about to integrate over.
• Reduce reference page — every domain and argument form, the engines behind them (sign diagrams, Fourier–Motzkin, CAD, the Diophantine reformatting), and their documented limits, in full.