Absolute Value Inequality

Definition

An absolute value inequality is a mathematical statement that involves the absolute value of a variable or expression being less than, greater than, or equal to a specific value. It is used to represent and solve problems where the distance of a quantity from a reference point is of interest.

5 Must Know Facts For Your Next Test

1. Absolute value inequalities can be solved using the same basic principles as linear inequalities, but with the added consideration of the absolute value function.
2. The solution set of an absolute value inequality can be represented as the union or intersection of two linear inequalities, depending on the type of inequality.
3. Graphing absolute value inequalities involves sketching the graph of the corresponding absolute value function and then shading the appropriate regions based on the inequality.
4. Absolute value inequalities can be used to model and solve real-world problems involving distance, time, and other quantities where the distance from a reference point is of interest.
5. The properties of absolute value, such as $|x| = |-x|$ and $|x| \geq 0$, are crucial in understanding and solving absolute value inequalities.

Review Questions

• Explain the relationship between absolute value and absolute value inequalities.
  • Absolute value is a fundamental concept that underlies absolute value inequalities. The absolute value of a number represents the distance of that number from zero on the number line, regardless of whether the number is positive or negative. This concept is then extended to inequalities, where the absolute value of a variable or expression is compared to a specific value. Solving absolute value inequalities involves understanding the properties of absolute value and applying the same principles used in solving linear inequalities, but with the added consideration of the absolute value function.
• Describe the process of graphing an absolute value inequality.
  • To graph an absolute value inequality, you first need to sketch the graph of the corresponding absolute value function. This involves drawing a V-shaped graph with the vertex at the point where the absolute value is zero. Then, based on the type of inequality (less than, greater than, or equal to), you need to shade the appropriate regions of the graph. For example, for the inequality $|x - 3| \leq 5$, you would shade the region between the two vertical lines at $x = -2$ and $x = 8$, as this represents the set of all values of $x$ whose distance from 3 is less than or equal to 5.
• Evaluate how absolute value inequalities can be used to model and solve real-world problems.
  • Absolute value inequalities can be a powerful tool for modeling and solving a variety of real-world problems. For instance, they can be used to represent situations where the distance from a reference point is of interest, such as travel time, delivery schedules, or safety zones. By setting up an absolute value inequality, you can determine the range of values that satisfy the given constraints, which can then be used to make informed decisions or optimize the solution. Additionally, the graphical representation of absolute value inequalities can provide a visual aid in understanding and interpreting the problem, making it easier to communicate and collaborate on real-world applications.

Related terms

Absolute Value: The absolute value of a number is the distance of that number from zero on the number line, regardless of whether the number is positive or negative.

Linear Inequality: A linear inequality is an inequality that involves a linear expression, such as $ax + b > c$, where $a$, $b$, and $c$ are constants.

Compound Inequality: A compound inequality is an inequality that combines two or more individual inequalities using the logical connectives 'and' or 'or'.