Name: solving nonlinear inequalities on a number line
Uploaded: 2025-11-26T05:43:48.000Z
Duration: 1 min 29 s
Channel: JoVE

공기 저항이 없는 수직 발사체 운동에서 물체는 초당 50미터의 속도로 지상에서 발사됩니다. 시간 경과에 따른 높이는 이차 방정식을 따릅니다: 음수 5 t 제곱 곱하기 50 t, 여기서 음수 5는 중력으로 인한 가속도의 절반입니다. 목표는 발사체의 높이가 100미터를 초과할 때의 시간 간격을 결정하는 것입니다.높이 함수는 2차이므로 이를 상수 값과 비교하려면 2차 부등식을 풀어야 합니다.부등식은 모든 항을 한쪽으로 이동하고 음의 5로 나누어 선행 계수를 1로 단순화하여 재배열됩니다.결과 표현식은 쉽게 인수분해할 수 없기 때문에 이차 공식이 사용됩니다. 계수는 두 개의 해점을 찾기 위해 공식에 대체됩니다.이 값은 발사체가 100미터 이상을 유지하는 기간의 시작과 끝을 나타냅니다.높이 함수의 그래프는 이러한 상황을 시각화하는 데 도움이 됩니다. 곡선은 100미터 표시 이상으로 올라가 정점에 도달한 다음 다시 표시로 떨어집니다.그래프에서 100미터 위에 있는 곡선 부분은 조건이 충족되는 시간 간격을 표시하기 위해 음영 처리됩니다.

비선형 부등식

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&#48708;&#49440;&#54805; &#48512;&#46321;&#49885;&#51008; &#51649;&#49440;&#51060; &#50500;&#45772; &#44257;&#49440; &#54805;&#53468;&#47196; ...

Education

교육

JoVE Core

Math Fundamentals

수학 기초

Unit

단위

Polynomial and Rational Functions

다항함수와 유리함수

KEY TERMS AND CONCEPTS

주요 용어 및 개념

Application of Nonlinear Inequalities

비선형 부등식의 적용

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 -2 split the number line into three regions. A sample value from each region is then tested in the expression to see whether that interval gives a positive or negative result.

The final answer includes only the regions that match the original sign of the inequality. If the inequality says the expression must be less than zero, then only the intervals with negative results are included. The same interval test works for rational inequalities, where the variable appears in the denominator. These problems also require checking where the expression is undefined, so any value that makes the denominator zero must be left out.

For example, a rational inequality may have key values of -1 and 4. These values divide the number line into separate intervals. Each interval is tested, but the point where the denominator becomes zero, such as 4, is excluded from the solution. This process is useful in real situations, such as finding when profit stays above zero, keeping machine pressure within safe limits, or checking when an object stays within a certain range of motion.

Solving Nonlinear Inequalities on a Number Line

solving-nonlinear-inequalities-on-a-number-line

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Quadratic Motion and Parabolas

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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,...

application of nonlinear inequalities

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 to solve nonlinear and rational inequalities by factoring, testing number line intervals, and excluding undefined values in grade 9–12 math.

Solving Nonlinear Inequalities on a Number Line | 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

Solving Nonlinear Inequalities on a Number Line

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