Name: absolute value inequalities on a number line
Uploaded: 2025-11-26T05:44:25.000Z
Duration: 1 min 23 s
Channel: JoVE

The absolute value indicates the distance a number lies from zero on the number line, regardless of whether the number is positive or negative.When the absolute value of a quantity is less than or equal to a given value, the solution includes all numbers between the negative and positive bounds.If the absolute value is greater than or equal to a given value, the solution includes all numbers outside those bounds.These solutions are shown on a number line using segments or rays, with circles indicating endpoints included because of the equality part of the inequality.Consider measuring the speed of sound using echoes from a known distance to a wall. Depending on environmental fluctuations, the measured speed is around 343 meters per second, with a variation of plus or minus 2 meters per second.This variation is modeled using an absolute value inequality, which sets a limit on how far the observed speed can deviate from the expected value.Solving the inequality involves subtracting and adding the allowed variation to the expected speed to get the full range of acceptable values.

absolute value inequalities

The absolute value is a mathematical tool that represents the distance of a number from zero on the number line, regardless of its sign. In the context of inequalities, absolute value expressions help define a range of permissible values or boundaries for a variable. These inequalities are commonly used in scientific modeling and data interpretation, where variability within or beyond a certain threshold must be captured precisely.An absolute value inequality of the form &#8739;x&#8739; &#8804; a, where a &#8805; 0, describes all values of x between &#8722;a and a, inclusive. This inequality can be rewritten as a compound inequality:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:38.9%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40036/40036_Equation_1_2__editor_40036.svg' alt='40036_Equation_1_2_.svg' /></figure>Graphically, this range is represented on a number line with a solid line segment connecting the endpoints at &#8722;a and a, often marked with closed circles to show that the endpoints are included in the solution.Conversely, an inequality of the form &#8739;x&#8739; &#8805; a indicates that x lies at a distance of at least a unit from zero. This solution set includes two separate intervals: &#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:37.63%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40036/40036_Equation_2_2__editor_40036.svg' alt='40036_Equation_2_2_.svg' /></figure>These are typically depicted using rays extending outward from the endpoints, often marked with closed circles if the endpoints are included.Consider a chemical reaction that proceeds optimally at 80&#8451;. To ensure safety and product yield, the temperature must remain within 2.5&#8451; of this value. This requirement can be expressed as:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:34.34%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40036/40036_Equation_3_2__editor_40036.svg' alt='40036_Equation_3_2_.svg' /></figure>This inequality means the temperature x can deviate no more than 2.5&#8451; from the ideal value. Solving the inequality gives:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:35.26%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40036/40036_Equation_4_2__editor_40036.svg' alt='40036_Equation_4_2_.svg' /></figure>Hence, any temperature within this interval maintains optimal reaction conditions.

Absolute Value Inequalities

The absolute value is a mathematical tool that represents the distance of a number from zero on the number line, regardless of its sign. In the context of ...

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absolute value inequalities on a number line

Absolute value inequalities show how far a number can be from zero or from a target value. Absolute value means distance on the number line, so the sign of the number does not matter. In inequalities, this distance helps define a valid range of values for a variable.

An inequality of the form |x| ≤ a, where a ≥ 0, includes every x between −a and a. This can also be written as a compound inequality. On a number line, the solution is shown as a solid segment from −a to a with closed circles at the endpoints, because the endpoints are part of the answer.

An inequality of the form |x| ≥ a means x is at least a units away from zero. Its solution has two separate intervals, one on each side of zero. These are usually drawn as rays extending outward from the endpoints, and closed circles are used when the endpoints are included.

These inequalities are useful in scientific modeling and data interpretation, where values must stay within a set limit or beyond a boundary. A chemical reaction that works best at 80°C can be used as an example. If the temperature must stay within 2.5°C of 80°C, the condition can be written as an absolute value inequality.

That inequality shows that the temperature x may vary only a small amount from the ideal value. Solving it gives an interval of acceptable temperatures. Any temperature in that interval keeps the reaction in its optimal range.

Absolute Value Inequalities on a Number Line

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Absolute value inequalities show how far a number can be from zero or from a target value. Absolute value means distance on the number line, so the sign of the number does not matter. In inequalities, this distance helps define a valid range of values for a variable. An inequality of the form |x| ≤ a, where a ≥ 0, includes every x between −a and a. This can also be written as a compound inequality. On a number line, the solution is shown as a solid segment from −a to a with closed circles at...

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key terms and concepts

Learn how absolute value inequalities show distance from zero, form compound inequalities, and appear on a number line with a science-based temperature example.

Absolute Value 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.

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Absolute Value Inequalities on a Number Line

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