Inequality Examples & Solutions Library — 59 Solved Problems
At a Glance
A comprehensive reference library of solved inequalities organized by type. Each example includes the solution interval and the key solving strategy. Use the interactive solver to verify any example step by step.
Introduction
Whether you are a student preparing for an exam or a teacher looking for curated examples, this library covers 59 solved inequalities across all major types. Every example shows the solution interval and the key strategy used to solve it.
💡 Tip: Click any example in our Inequality Solver to solve it interactively with full step-by-step explanation.
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🔵 Linear Inequalities
Linear inequalities take the form ax + b < c. The key rule: flip the sign when multiplying or dividing by a negative number.
| Inequality | Solution | Key Strategy |
|---|---|---|
3x + 2 < 11 | x ∈ (-∞, 3) | Isolate x |
-3x + 1 > -8 | x ∈ (-∞, 3) | Dividing by a negative flips the sign |
(x+1)/2 >= 3 | x ∈ [5, ∞) | Multiply both sides by 2 (positive) |
5 - 2x <= 1 | x ∈ [2, ∞) | Move x terms to one side |
4x - 7 > 2x + 3 | x ∈ (5, ∞) | Collect x terms |
-x/3 < 4 | x ∈ (-12, ∞) | Multiply by -3 (flip sign) |
→ Solve linear inequalities instantly: Inequality Solver with steps
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🟣 Quadratic Inequalities
Quadratic inequalities require finding roots and analyzing the sign of the parabola between and outside those roots.
| Inequality | Solution | Key Strategy |
|---|---|---|
x^2 - 5x + 6 < 0 | x ∈ (2, 3) | Factor: (x-2)(x-3) < 0 |
x^2 - 4 > 0 | x ∈ (-∞,-2) ∪ (2,∞) | Difference of squares |
x^2 + 2x <= 8 | x ∈ [-4, 2] | Complete the square |
2x^2 - 3x - 2 >= 0 | x ∈ (-∞,-1/2] ∪ [2,∞) | Quadratic formula |
-x^2 + 1 > 0 | x ∈ (-1, 1) | Negative leading coefficient |
x^2 + 1 < 0 | ∅ (no solution) | Discriminant < 0 |
x^2 - 6x + 9 <= 0 | x = {3} | Perfect square |
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🟤 Higher Degree Inequalities
For polynomials of degree ≥ 3, use the sign chart method after factoring.
| Inequality | Solution | Key Strategy |
|---|---|---|
x^3 - x > 0 | x ∈ (-1,0) ∪ (1,∞) | Factor x(x-1)(x+1), sign chart |
x^3 - 4x^2 + 4x < 0 | x ∈ (-∞, 0) | Factor x(x-2)², sign chart |
x^4 - 5x^2 + 4 <= 0 | x ∈ [-2,-1] ∪ [1,2] | Substitution u = x² |
x^3 + 2x^2 - x - 2 > 0 | x ∈ (-2,-1) ∪ (1,∞) | Factor by grouping |
x^3 >= 8 | x ∈ [2, ∞) | Cube root both sides |
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🟠 Rational Inequalities
Rational inequalities require finding zeros of numerator and denominator, then analyzing sign changes. Never cross-multiply without checking the sign.
| Inequality | Solution | Key Strategy |
|---|---|---|
(x-1)/(x+2) < 0 | x ∈ (-2, 1) | Sign chart with critical points x=1, x=-2 |
(x+3)/(x-1) >= 2 | x ∈ (1, 5] | Bring to one side, common denominator |
1/x > 2 | x ∈ (0, 1/2) | Split by sign of x |
(x^2-4)/(x-1) <= 0 | x ∈ (-∞,-2] ∪ (1,2] | Factor numerator |
(2x+1)/(x^2+1) > 0 | x ∈ (-1/2, ∞) | Denominator always positive |
x/(x+1) < 1/(x-1) | x ∈ (-1, 1) | Common denominator, sign chart |
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🔴 Absolute Value Inequalities
Two cases: |f(x)| < a → -a < f(x) < a (intersection), and |f(x)| > a → f(x) < -a OR f(x) > a (union).
| Inequality | Solution | Key Strategy |
|---|---|---|
abs(x) < 3 | x ∈ (-3, 3) | Two-case split |
abs(2x - 1) >= 5 | x ∈ (-∞,-2] ∪ [3,∞) | Two-case split |
abs(x + 2) <= 4 | x ∈ [-6, 2] | Split into double inequality |
abs(3x - 2) > 1 | x ∈ (-∞, 1/3) ∪ (1,∞) | Union of two intervals |
abs(x^2 - 4) < 5 | x ∈ (-3, 3) | Split, solve quadratics |
abs(1/x) > 2 | x ∈ (-1/2, 0) ∪ (0, 1/2) | Split by sign of x |
abs(x - 1) + abs(x + 1) < 4 | x ∈ (-2, 2) | Case analysis on sign |
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🟢 Radical Inequalities
For radical inequalities, isolate the radical, then square both sides only when both sides are non-negative. Always check the domain.
| Inequality | Solution | Key Strategy |
|---|---|---|
sqrt(x) < 3 | x ∈ [0, 9) | Square both sides (x≥0) |
sqrt(x - 2) >= 1 | x ∈ [3, ∞) | Isolate radical, square |
sqrt(2x + 1) < x | x ∈ (3, ∞) | Square, check domain |
sqrt(x^2 - 1) > x | x ∈ (-∞, -1) | Domain + case analysis |
sqrt(x + 3) + sqrt(x) > 3 | x ∈ (1, ∞) | Isolate one radical, square twice |
sqrt(x) <= x - 2 | x ∈ [4, ∞) | Square, check non-negativity |
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🔵 Trigonometric Inequalities
Trigonometric inequalities use the unit circle to find solution intervals over specified domains.
| Inequality | Solution (0 ≤ x < 2π) | Key Strategy |
|---|---|---|
sin(x) > 1/2 | x ∈ (π/6, 5π/6) | Unit circle, reference angle |
cos(x) <= 0 | x ∈ [π/2, 3π/2] | Unit circle, second/third quadrant |
tan(x) > 1 | x ∈ (π/4, π/2) ∪ (5π/4, 3π/2) | Periodicity π |
2sin(x) - 1 >= 0 | x ∈ [π/6, 5π/6] | Isolate sin(x) |
cos(2x) < 1/2 | x ∈ (π/6, 5π/6) ∪ (7π/6, 11π/6) | Double angle substitution |
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🔵 Exponential & Logarithmic Inequalities
Remember: log is increasing (preserve inequality) for base > 1; decreasing (flip inequality) for 0 < base < 1.
| Inequality | Solution | Key Strategy |
|---|---|---|
2^x > 8 | x ∈ (3, ∞) | Same base: x > 3 |
e^x <= 1 | x ∈ (-∞, 0] | ln of both sides |
log(x) > 2 | x ∈ (100, ∞) | Base 10, exponentiate |
ln(x - 1) < 3 | x ∈ (1, e³+1) | Exponentiate, check domain |
3^(2x-1) >= 27 | x ∈ [2, ∞) | Same base: 2x-1 ≥ 3 |
log_2(x^2 - 3) > 2 | x ∈ (-∞,-√7) ∪ (√7,∞) | Exponentiate, solve quadratic |
0.5^x < 0.25 | x ∈ (2, ∞) | Base < 1 flips inequality |
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🟡 Application Problems
Real-world inequalities from physics, economics, and optimization.
| Inequality | Solution | Key Strategy |
|---|---|---|
v^2 <= 2*a*d (kinematics) | v ∈ [-√(2ad), √(2ad)] | Square root both sides |
P*r*t <= I_max (interest) | t ∈ [0, I_max/(Pr)] | Divide by positive Pr |
C(x) <= R(x) profit | x ∈ [x₁, x₂] | Break-even analysis |
Explore more real-world math: Integral Calculator · Derivative Calculator · Limit Calculator
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⚠️ Special Cases
| Inequality | Solution | Why |
|---|---|---|
x^2 + 1 < 0 | ∅ | No real solution (always positive) |
x^2 >= 0 | ℝ (all reals) | Always true |
(x-1)^2 < 0 | ∅ | Square is always ≥ 0 |
x^2 - 2x + 1 <= 0 | x = {1} | Perfect square = 0 only |
abs(x) < 0 | ∅ | Absolute value is never negative |
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How to Solve Any Inequality — 5 Steps
- Move everything to one side (make RHS = 0)
- Find critical points (roots of numerator and denominator)
- Draw a sign chart — test each interval
- Apply domain restrictions (for radical/log/trig)
- Write the solution as an interval using ∪ and ∩
→ Try it now with our Inequality Solver with steps — get instant solutions with a full sign-chart breakdown.
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Related Tools on Mathsway
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| Equation Solver | Algebraic equations (polynomial, rational, trigonometric) |
| Derivative Calculator | Finding f'(x) step by step |
| Integral Calculator | Definite and indefinite integrals |
| Limit Calculator | Limits, L'Hôpital's rule |