Name: graphing and solving nonlinear inequalities
Uploaded: 2025-11-26T05:43:49.000Z
Duration: 1 min 25 s
Channel: JoVE

在线性不等式中，边界是一条直线，用于分隔解区域和满足不等式其他条件的区域。相比之下，非线性不等式包含高次表达式，并使用曲线边界而非直线来定义区域。在处理此类不等式（尤其是多项式类型）时，通常通过因式分解表达式，并将每个因式设为零来求解根（或称零点）。这些点不包含在解集中，但表示表达式变号的位置，从而将数轴划分为若干区间，可在各区间内检验不等式的成立情况。从每个区间中选取一个值代入原不等式，以确定该表达式在该区域内是否成立。不等式成立的区间即为解区域。在商业背景下，利润是收入与已售出产品成本之间的差额。为确保至少20美元的利润，该条件可表示为一个非线性不等式。求解此不等式可得到满足利润要求的具体产品数量范围。

非线性不等式简介

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Education

教育

JoVE Core

Math Fundamentals

数学基础

Unit

单位

Polynomial and Rational Functions

多项式与有理函数

KEY TERMS AND CONCEPTS

关键术语与概念

Introduction to Nonlinear Inequalities

graphing and solving nonlinear inequalities

Linear and nonlinear inequalities help describe variable relationships and the ranges that satisfy a condition. A linear inequality uses variables raised only to the first power. Its graph is a straight line that divides the coordinate plane into two regions. One region satisfies the inequality, and the other does not.

Nonlinear inequalities use variables raised to powers greater than one. These include quadratic inequalities and higher-degree polynomial inequalities. Their graphs are curved, not straight. The changing shape shows that the rate of change is not constant as the input changes.

Solving a nonlinear inequality often means looking at intervals on the number line. Critical points are the values where the expression equals zero. These points show where the function may change sign. They split the number line into sections that can be tested with a representative value from each interval.

The intervals that make the inequality true form the solution set. This approach is useful in applied settings such as economics and business. A profit function can depend on production volume and marginal costs. By using a nonlinear inequality, it is possible to find the sales range that keeps profit above a target such as five hundred dollars.

Graphing and Solving Nonlinear Inequalities

graphing-and-solving-nonlinear-inequalities

quadratic models

Quadratic Motion and Parabolas

Quadratic models describe relationships where the rate of change changes at a constant rate. In math and science, this often shows up as a curved pattern called a parabola. These models help explain how a system changes over time or under a set force. A common example is the vertical motion of an object moving under gravity, such as a ball thrown into the air. The object rises to a maximum height and then falls back down. Gravity slows the ascent and speeds up the descent, which creates the...

introduction to polynomial functions

Polynomial Graphs, Zeros, and Degree

Polynomial functions are an important part of algebra and calculus. They are made from variables and constants combined by addition, subtraction, and multiplication. The variable is raised to a nonnegative integer exponent. A general polynomial function of degree n has a leading term aₙxⁿ, where aₙ is not equal to 0. The coefficient aₙ is the leading coefficient, and a₀ is the constant term. Polynomials are grouped by degree. A constant function has degree zero, a linear function has degree...

long division of polynomials

Polynomial Division: Quotient and Remainder

Polynomial division breaks a polynomial into a quotient and a remainder. It works like numerical division, but the dividend is a polynomial and the divisor is an algebraic expression. The result shows how much of the divisor fits into the dividend and what is left over. The first step is to write both polynomials in standard form, with terms in descending powers of the variable. Then divide the leading term of the dividend by the leading term of the divisor. That first result becomes the first...

synthetic disvision of polynomials

Using Synthetic Division to Find Factors

Synthetic division is a fast way to divide a polynomial by a linear binomial of the form x - c, where c is a real number. It gives a shorter method than polynomial long division and is useful for checking values and finding factors. The setup starts with the coefficients of the polynomial written in descending order of powers. If a term is missing, a zero must be used to keep the columns lined up. The constant c from the divisor x - c is placed to the left of the synthetic division table. The...

real zeros of polynomials

Finding Polynomial X-Intercepts

Polynomials can be analyzed by finding their real zeros, which are the values of x that make the polynomial equal to zero. These points are also the x-intercepts of the graph. Understanding them helps students see where a polynomial crosses the x-axis. The Rational Zeros Theorem gives a list of possible rational solutions for a polynomial with integer coefficients. If f(x)=anx^n+...+a0, then every rational zero has the form p/q. In that form, p divides the constant term a0 and q divides the...

fundamental theorem of algebra

Polynomial Roots in Complex Numbers

Polynomial roots in complex numbers are a key result in algebra. The Fundamental Theorem of Algebra says that every non-constant polynomial with complex coefficients has at least one complex zero. For a polynomial of degree n, where n is at least 1 and the leading coefficient a n is not 0, there is at least one solution in the complex number system. This result also applies to polynomials with real coefficients, because the real numbers are a subset of the complex numbers. The Complete...

complex zeros

Finding Polynomial Roots With Complex Numbers

Complex numbers help describe every root of a polynomial equation. A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit, with i^2 = -1. A complex zero is any value that makes P(x) = 0 for a polynomial P(x). The complex number system includes the real numbers, so it gives a complete way to study polynomial roots. Every polynomial of degree n  1 can be written as a product of n linear factors over the complex numbers. In that factorization, each c_i...

asymptotes in rational functions

Graphing Rational Functions with Asymptotes

Rational functions are quotients of two polynomials, with the denominator Q(x) not equal to zero. Their graphs often include asymptotes, which are lines the graph gets close to but does not touch. These lines help show how the function behaves near certain input values and at large values of x. A vertical asymptote appears when the denominator is zero and the numerator is not zero. At those x-values, the function is undefined. To find a vertical asymptote, solve Q(x) = 0. For example, a...

introduction to nonlinear inequalities

Linear and nonlinear inequalities help describe variable relationships and the ranges that satisfy a condition. A linear inequality uses variables raised only to the first power. Its graph is a straight line that divides the coordinate plane into two regions. One region satisfies the inequality, and the other does not. Nonlinear inequalities use variables raised to powers greater than one. These include quadratic inequalities and higher-degree polynomial inequalities. Their graphs are curved,...

application of nonlinear inequalities

Solving Nonlinear Inequalities on a Number Line

Nonlinear inequalities can be solved by checking where an expression is positive or negative on a number line. These inequalities often include squares, products, or variables in the denominator, so they do not follow the same simple pattern as linear inequalities. The first step is to rewrite the inequality so that zero is on one side. After rewriting, factor the expression if possible. Factoring helps you find the key values where the expression equals zero. In one example, the values 3 and...

key terms and concepts

Learn how linear and nonlinear inequalities differ, how critical points split a number line, and how interval testing finds the solution set.

Graphing and Solving Nonlinear Inequalities | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Graphing and Solving Nonlinear Inequalities

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