Solve a linear inequality and display the solution set using interval notation
a =
b =
Inequality
c =
Result
Inequality
Solution Set
Step-by-Step Derivation
Inequality Solving Formula
ax+b < c → ax < c-b → x < (c-b)/a (a>0) ax+b < c → ax < c-b → x > (c-b)/a (a<0, inequality sign reverses)
The key steps to solving linear inequalities: first move terms, then divide by the coefficient of the variable. Remember: when the coefficient is negative, the inequality sign must be reversed.
⚠Note: Special handling is needed when a=0. If the inequality is always true, the solution is ℝ. If contradictory, there is no solution. In interval notation, ( and ) denote excluded endpoints, [ and ] denote included endpoints.
What Is Inequality Solving?
Solving inequalities means finding the range of values for the unknown variable that makes the inequality true. Unlike equations, the solution of an inequality is one or more continuous intervals.
Transposition Preserves Sign
Moving terms from one side to another does not change the equality property, and the inequality direction remains unchanged.
Flip Sign When Dividing by Negative
When both sides are divided by a negative number, the inequality direction must be reversed. For example, -2x < 6 → x > -3.
Interval Notation
(a,∞) open interval excludes a; [a,∞) closed interval includes a; (-∞,b) excludes b; (-∞,b] includes b.
Solution Set Properties
The solution set of an inequality is an interval (possibly no solution x∈∅, or all real numbers x∈ℝ).
💡 Teaching Example: Solve 2x-4 < 0. Transpose: 2x < 4. Divide both sides by 2 (positive, sign unchanged): x < 2. Solution set: (-∞, 2).
What is the difference between solving inequalities and equations?▼
Equations usually yield a finite set of specific numbers, while inequalities give one or more continuous intervals. Also, when dividing by a negative number, the inequality sign must be reversed.
Why does dividing by a negative flip the inequality sign?▼
For example, -2 < -1 is true. Dividing both sides by -1 gives 2 > 1, which is still true but the sign is reversed. This is because dividing by a negative number reverses the order relationship.
How do you express solution sets in interval notation?▼
(a, b) is an open interval (excludes a and b); [a, b] is a closed interval (includes a and b). x > 2 is written as (2, ∞), x ≤ 5 as (-∞, 5].
How to solve quadratic inequalities?▼
First find the roots x₁ and x₂ of the corresponding quadratic equation (x₁ < x₂). If a > 0, then x²+bx+c > 0 has solution (-∞, x₁)∪(x₂, ∞) (outside the roots), and x²+bx+c < 0 has solution (x₁, x₂) (between the roots). If a < 0, the direction is reversed.
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.