Linear Inequalities
Solve ax+b<c or ax+b>c like equations, but flip the inequality when multiplying or dividing by a negative. Compound a<bx+c<d: solve for x in the middle. |x−a|<b → (a−b,a+b).
Did our AI summary help? Let us know.
Dividing by negative: −2x>6 → x<−3 (flip!). Compound: solve each part, solution = intersection. |x−a|<b gives open interval (a−b,a+b).
Ready to run the numbers?
Why: Inequalities model constraints: budget limits, tolerance bands, feasibility. Critical rule: multiplying or dividing by a negative flips the inequality. Solution is usually an interval.
How: Isolate x like an equation. If you multiply or divide by a negative, flip < to > and vice versa. Compound: solve all parts, intersect. |x−a|<b: −b<x−a<b → a−b<x<a+b.
Run the calculator when you are ready.
Linear Inequality Solver
Solve ax+b op c, compound a<bx+c<d, |x−a|<b. Interval notation, set-builder, steps.
Boundary & Test Points
Solution vs Excluded
Steps
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
−3x<9 → x>−3. Must flip when dividing by −3.
— Sign flip
|x−5|<2 → 3<x<7. Distance from 5 less than 2.
— Absolute
Key Takeaways
- • Sign flip: When multiplying or dividing both sides by a negative number, reverse the inequality (< ↔ >, ≤ ↔ ≥).
- • Interval notation: ( ) = open (excluded), [ ] = closed (included). (−∞, a) means x < a.
- • Compound inequalities: a < bx + c < d means solve both parts and take the intersection.
- • Absolute value: |x − a| < b ⟺ a − b < x < a + b.
- • Set-builder: {x ∈ ℝ : condition} describes the solution set explicitly.
- • Real-world: Budgets, temperature ranges, speed limits often use linear inequalities.
Did You Know?
How It Works
To solve ax + b < c: (1) Subtract b from both sides: ax < c − b. (2) Divide by a. If a < 0, flip the inequality. (3) Express in interval notation. For compound a < bx + c < d, solve the middle part for x and ensure both outer inequalities hold. For |x−a| < b, use the equivalence a−b < x < a+b.
Sign Flip Reference
Divide by negative: < → >, > → <, ≤ → ≥, ≥ → ≤
Expert Tips
Always Check the Coefficient
Before dividing, check if the coefficient of x is negative. That is the only time you flip the inequality.
Test a Value
Pick a number in your solution set and plug it into the original inequality to verify it works.
Compound = Intersection
a < x < b means x must satisfy BOTH x > a AND x < b. It is the intersection of two half-lines.
Interval Endpoints
Use ( ) for strict < or >; use [ ] for ≤ or ≥. Infinity always gets parentheses.
Inequality to Interval Reference
| Inequality | Interval | Set-builder |
|---|---|---|
| x < a | (-∞, a) | {x ∈ ℝ : x < a} |
| x ≤ a | (-∞, a] | {x ∈ ℝ : x ≤ a} |
| x > a | (a, ∞) | {x ∈ ℝ : x > a} |
| x ≥ a | [a, ∞) | {x ∈ ℝ : x ≥ a} |
| a < x < b | (a, b) | {x ∈ ℝ : a < x < b} |
| a ≤ x ≤ b | [a, b] | {x ∈ ℝ : a ≤ x ≤ b} |
FAQ
Why do we flip the inequality when dividing by a negative?
Because multiplying/dividing by a negative reverses order: if a < b, then −a > −b. So −2x ≥ 6 becomes x ≤ −3.
What is interval notation?
A compact way to write solution sets. (a,b) means a < x < b; [a,b] means a ≤ x ≤ b. Parentheses = excluded, brackets = included.
What is a compound inequality?
Two inequalities combined: a < bx + c < d. Solve for x so both parts hold. Result is an interval (a′, b′).
How do I solve |x − 3| < 5?
Equivalent to −5 < x − 3 < 5, so −2 < x < 8. Interval: (−2, 8).
When is the solution "all real numbers"?
When you get a true constant inequality (e.g., 0 < 5) or when the variable cancels and the remaining statement is always true.
When is there no solution?
When you get a false constant (e.g., 5 < 2) or when the variable cancels and the statement is always false.
Quick Reference
Disclaimer: This calculator is for educational use. Verify solutions by testing sample values. Not a substitute for formal mathematical instruction.
Related Calculators
Absolute Value Inequalities Calculator
Solve absolute value inequalities |ax+b| < c, > c, ≤ c, ≥ c. Get solution intervals, interval notation, set builder notation, number line visualization...
MathematicsAbsolute Value Equation Calculator
Solve absolute value equations and inequalities including |ax+b|=c, |ax+b|=|cx+d|, and |ax+b|≤c forms. Features solution verification, number line...
MathematicsGraphing Quadratic Inequalities Calculator
Solve and interpret quadratic inequalities ax²+bx+c < 0, > 0, ≤ 0, ≥ 0. Get solution intervals, roots, vertex, discriminant, sign chart, interval notation...
MathematicsInterval Notation Calculator
Convert between interval notation, inequality notation, and set-builder notation. Support for union (∪), intersection (∩), complement, and absolute value |x−a|<b. Unbounded intervals (−∞, a) and (a, ∞). Open vs closed endpoints. Number line representation and interval length. Step-by-step solutions and De Morgan's laws.
MathematicsBessel Function Calculator
Calculate Bessel functions J_n(x), Y_n(x), I_n(x), and K_n(x) with series approximation. Supports first kind (J), second kind/Neumann (Y), and modified Bessel functions (I, K). Applications include vibrating drum modes, electromagnetic waveguides, heat conduction in cylinders, and Bessel filter design in signal processing. Uses the power series J_n(x) = Σ (-1)^m/(m!(m+n)!)·(x/2)^(2m+n). Includes convergence info, Bar chart comparing values at different x, Doughnut chart showing series term contributions, and educational content on Bessel's differential equation and cylindrical harmonics.
MathematicsBinomial Coefficient Calculator
Calculate binomial coefficients C(n,k) = n!/(k!(n-k)!) — the number of ways to choose k items from n without regard to order. Also known as n choose k or ⁿCₖ. Features Pascal's triangle row generation, factorial breakdown, Bar chart of full row C(n,0)..C(n,n), Doughnut chart showing k vs n−k. Applications: lottery (C(49,6)), poker hands (C(52,5)), committee selection, DNA combinations, binary strings. Educational content on combinatorics, binomial theorem (x+y)ⁿ expansion, and Pascal's identity.
Mathematics