What happens when you multiply or divide an inequality by a negative number?

When you multiply or divide both sides of an inequality by a negative number, you must flip (reverse) the inequality sign. For example, -2x > 6 becomes x < -3 after dividing by -2. This is one of the most common errors when solving inequalities. The rule applies regardless of whether the negative number is an integer, fraction, or variable — but for variables you must consider cases based on the variable's sign.

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Inequality Solver

Linear · Compound · Quadratic · Rational · Polynomial

Solve any inequality step by step. Get interval notation, sign chart, and a number line diagram — instantly.

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Form: ax + b OP c (e.g. 2x + 3 > 7)

Relation

What Is an Inequality?

An inequality is a mathematical statement that compares two expressions using one of the relational operators: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike equations, inequalities typically have infinitely many solutions, described as an interval or union of intervals on the number line.

Inequalities appear everywhere in real life: a budget constraint says your spending must be less than or equal to your income; a speed limit says your speed must be less than or equal to the posted limit; an engineering tolerance specifies a part's dimension must stay within a range. Mastering inequality solving is essential for algebra, calculus, and applied mathematics.

Types of Inequalities

Linear Inequalities

A linear inequality has the form ax + b > c (or with any other relational operator). The solution is always a single ray on the number line. Solve by isolating x exactly as you would in a linear equation, remembering to flip the inequality sign when multiplying or dividing by a negative number.

Compound (Two-Sided) Inequalities

A compound inequality such as a < bx + c ≤ d places two constraints on x simultaneously. Solve by performing identical operations on all three parts. The solution is a bounded interval, either open, closed, or half-open depending on the strictness of each side.

Quadratic Inequalities

A quadratic inequality has the form ax² + bx + c > 0. The key steps are: find the roots of the corresponding quadratic equation, use those roots to divide the number line into intervals, and test a point in each interval to determine the sign. The discriminant D = b² − 4ac determines the number of real roots and shapes the solution.

Discriminant	Roots	ax²+bx+c > 0 solution (a>0)
D > 0	Two distinct real roots r₁, r₂	(-∞, r₁) ∪ (r₂, +∞)
D = 0	One repeated root r	(-∞, r) ∪ (r, +∞)
D < 0	No real roots	All real numbers (-∞, +∞)

Rational Inequalities

A rational inequality involves a fraction with x in the denominator, such as (ax+b)/(cx+d) > 0. Critical points come from both the numerator roots (where the expression equals zero) and denominator roots (where it is undefined and must be excluded). Build a sign chart across all intervals to determine where the fraction is positive or negative.

Polynomial Inequalities

A polynomial inequality of degree 3 or higher is solved by finding all real roots, building a sign chart, and determining which intervals satisfy the inequality. The sign alternates between consecutive roots for polynomials with single roots, and stays the same across a repeated root.

Interval Notation Guide

( ) parenthesis — endpoint is excluded (strict inequality < or >)
[ ] bracket — endpoint is included (non-strict ≤ or ≥)
∞ always uses a parenthesis — infinity is never reached
∪ union — joins two or more disjoint solution intervals
∅ empty set — the inequality has no solution
(-∞, +∞) — all real numbers

Real-World Applications

Budget constraints: If you earn $50/hr and need at least $200 for the week, 50h ≥ 200 means h ≥ 4 hours.
Speed limits: A car must stay within 0 ≤ v ≤ 60 mph on a highway — a compound inequality.
Engineering tolerances: A bolt's diameter must satisfy |d − 10| ≤ 0.05 mm, an absolute-value (compound) inequality.
Profit analysis: Profit = −x² + 100x − 1200 > 0 determines the production range where a business is profitable.
Physics (projectile range): −16t² + 64t > 0 gives the time interval when a projectile is above the ground.

Frequently Asked Questions

What is interval notation and how do I read it?

Interval notation uses parentheses and brackets to describe a set of numbers. A parenthesis means the endpoint is excluded (open), while a bracket means it is included (closed). For example, (2, 5] means all x where 2 < x ≤ 5. Infinity always uses parentheses. The union symbol ∪ joins disjoint intervals such as (−∞, 1) ∪ (3, +∞).

How do you solve a quadratic inequality like x²−5x+6 < 0?

Find the roots: x²−5x+6=0 factors as (x−2)(x−3)=0, giving roots x=2 and x=3. Since the parabola opens upward (a=1>0), it is negative between the roots. So x²−5x+6 < 0 for x ∈ (2, 3). The endpoints are excluded because the inequality is strict (<), not ≤.

What is a sign chart and when do I need one?

A sign chart shows the sign (+, −, or 0) of a function in each interval between critical points. You need one when solving quadratic, rational, or polynomial inequalities. List the critical points (roots of numerator and denominator), test a value in each interval, and read off which intervals satisfy the inequality.

How do you solve a rational inequality like (x−1)/(x+2) > 0?

Find critical points: numerator root x=1 and denominator root x=−2 (excluded). Test intervals: x<−2 (try −3): (−4)/(−1)=4>0 ✓; −2<x<1 (try 0): (−1)/(2)<0 ✗; x>1 (try 2): (1)/(4)>0 ✓. Solution: (−∞,−2) ∪ (1,+∞). Note x=−2 is always excluded.

What happens when you multiply an inequality by a negative number?

The inequality sign must be flipped (reversed). For example, −2x > 6 becomes x < −3 after dividing by −2. This is one of the most common errors. The rule applies for any negative number — integer, fraction, or decimal.

How do you solve a compound inequality like −3 < 2x−1 ≤ 5?

Perform the same operation on all three parts simultaneously. Add 1: −2 < 2x ≤ 6. Divide by 2: −1 < x ≤ 3. Solution: (−1, 3]. The left endpoint is open (strict <), and the right endpoint is closed (≤).

Can a quadratic inequality have no solution or all real numbers as the solution?

Yes. For ax² + bx + c > 0 with discriminant D = b²−4ac: if D < 0 and a > 0, the parabola is always positive, so the solution is all real numbers. If D < 0 and a < 0, the parabola is always negative, so there is no solution (∅). If D = 0, there is one repeated root and the parabola touches (but does not cross) the x-axis.

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