≥Inequalities Calculator

Linear, polynomial, rational, absolute value, quadratic, systems — plus calculus inequalities: increasing/decreasing intervals and concavity — all with step-by-step sign charts and interval notation.

Linear: ax + b ▷ c
a = b = c =
Flip rule: dividing/multiplying by negative reverses the inequality direction.
Steps: Full working
Classic Examples
Linear2x−1 > 5Linear−3x ≤ 9Linearx/2+1 ≥ 4Polyx²−4 > 0Polyx³−x < 0Polyx²+x−6 ≤ 0Polyx⁴−5x² ≥ 0Rational(x−2)/(x+1) > 0Rational(x+3)/(x−1) ≤ 0Abs Val|x| < 3Abs Val|2x−1| ≥ 5Abs Val|x+2| ≤ 4Quadraticx²−5x+6 < 0Quadraticx²+2x−8 ≥ 0Quadratic2x²−x−3 ≤ 0Increasingx³−3x on [−3,3]Increasingxe^−x on [−1,5]Concavityx³ on [−2,2]Concavitysin(x) on [−2π,2π]
Steps Modes
None  Answer only (interval notation).
Summary  Method pills.
Detailed  Full accordion: each algebraic step, sign chart, interval selection.
Operations Guide
ax — Linear
Isolate x; flip sign when dividing by negative
p(x) — Polynomial
Sign chart: roots divide number line into intervals
p/q — Rational
Zeros of p AND zeros of q are critical points; q-zeros excluded
|f| — Absolute Value
|f|k ↔ f<−k or f>k
ax² — Quadratic
Discriminant + parabola direction (∪ or ∩)
2D — System
Feasible region = intersection of half-planes
f' — Increasing/Dec
f'>0 ↑, f'<0 ↓, critical pts where f'=0
f'' — Concavity
f''>0 ∪, f''<0 ∩, inflection where f'' changes sign
Key Rules & Theorems
Flip Rule \(a<0\Rightarrow ax>b\Leftrightarrow x
Sign Chart \(\text{roots divide }\mathbb{R}\text{ into intervals}\)
Abs Value < \(|f|
Abs Value > \(|f|>k\Leftrightarrow f<-k\text{ or }f>k\)
Quadratic Δ \(\Delta>0\text{: 2 roots, }=0\text{: 1, }<0\text{: none}\)
Parabola ∪ \(a>0\Rightarrow f>0\text{ outside roots}\)
Parabola ∩ \(a<0\Rightarrow f>0\text{ between roots}\)
Rational Rule \(q(x)=0\text{ always excluded}\)
Increasing \(f'(x)>0\Rightarrow f\text{ increasing}\)
Decreasing \(f'(x)<0\Rightarrow f\text{ decreasing}\)
Concave Up \(f''(x)>0\Rightarrow\cup\)
Concave Down \(f''(x)<0\Rightarrow\cap\)
Inflection \(f''\text{ changes sign at }c\)
EVT \(f\text{ cont on }[a,b]\Rightarrow\exists\min,\max\)
IVT \(f\text{ cont, }f(a)f(b)<0\Rightarrow\exists\text{ root}\)
Input Tips
  • Use ^ for powers, * for multiply
  • Polynomial/Rational: move all terms to one side first (vs 0)
  • Absolute Value: enter only the expression inside |·|
  • Increasing/Concavity: f'(x) and f''(x) computed numerically
  • Search bounds for polynomial/calculus ops: use [−10, 10] or wider
  • Roots found by bisection on 600 sub-intervals
  • All results in interval notation (or feasible region description)
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