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

In a vertical projectile motion without air resistance, an object is launched from ground level with a velocity of 50 meters per second. Its height over time follows the quadratic equation: negative five t squared plus fifty t, where negative five is half the acceleration due to gravity. The goal is to determine the time interval when the projectile&#8217;s height exceeds 100 meters.Since the height function is quadratic, comparing it to a constant value involves solving a quadratic inequality.The inequality is rearranged by moving all terms to one side and dividing by negative five, which simplifies the leading coefficient to one.Since the resulting expression cannot be factored easily, the quadratic formula is used. The coefficients are substituted into the formula to find two solution points.These values represent the beginning and end of the time period during which the projectile stays above 100 meters.A graph of the height function helps visualize this situation. The curve rises above the 100-meter mark, reaches a peak, and then falls back to the mark.On the graph, the part of the curve that lies above 100 meters is shaded to show the time interval during which the condition is met.

application of nonlinear inequalities

A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality:&#160;&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:80.32%;'><img style='aspect-ratio:1920/74;' src='https://cloudfront.jove.com/files/ftp_upload/40060/40060_Equation_1_editor_40060.svg' alt='40060_Equation_1.svg' /></figure>can be factored as:&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:100%;'><img style='aspect-ratio:300/9;' src='https://cloudfront.jove.com/files/ftp_upload/40060/40060_Equation_2_editor_40060.svg' alt='40060_Equation_2.svg' /></figure>This form makes it easier to identify the values that cause the expression to equal zero. In this case, the key values are 3 and -2. These values divide the number line into three regions.The solver chooses one sample value in each region and substitutes it into the expression. This test shows whether the expression within that section is positive or negative. If the original inequality requires the expression to be less than zero, the correct regions are those where the result is negative. The complete solution includes only those regions.The same approach applies to rational inequalities, where the variable appears in the denominator. However, the solver must also consider where the expression is undefined. For example:&#160;&#160;&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:97.83%;'><img style='aspect-ratio:300/20;' src='https://cloudfront.jove.com/files/ftp_upload/40060/40060_Equation_3_editor_40060.svg' alt='40060_Equation_3.svg' /></figure>In this case, the key values are -1 and 4. These divide the number line into separate intervals. The solver tests each interval and excludes the point where the denominator becomes zero&#8212;in this case, 4.Nonlinear inequalities are useful in practical settings. A business analyst might use one to determine when profit stays above zero. An engineer might apply one to ensure a machine operates within safe pressure limits. A physicist might analyze when an object stays within a certain range of motion.

Application of Nonlinear Inequalities

A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often ...

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Math Fundamentals

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Polynomial and Rational Functions

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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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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...

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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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