An absolute value inequality is a mathematical expression that involves the absolute value of a variable or expression and a comparison operator such as less than, greater than, less than or equal to, or greater than or equal to. These inequalities are used to describe a range of values that satisfy the given condition.
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Absolute value inequalities can be used to describe the range of values that satisfy a given condition, such as the distance of a variable from a specific value.
Solving absolute value inequalities involves isolating the absolute value expression and then considering two cases: one where the expression is greater than or equal to zero, and one where the expression is less than zero.
The solution to an absolute value inequality can be represented using interval notation, which describes the range of values that satisfy the inequality.
Absolute value inequalities can be used to model real-world situations, such as describing the acceptable range of temperatures for a particular process or the distance from a target value.
Graphing absolute value inequalities on a number line can help visualize the solution set and understand the range of values that satisfy the inequality.
Review Questions
Explain the process of solving an absolute value inequality, and provide an example to illustrate the steps.
To solve an absolute value inequality, the general process is as follows:
1. Isolate the absolute value expression on one side of the inequality.
2. Consider two cases: one where the expression inside the absolute value is greater than or equal to 0, and one where the expression is less than 0.
3. Solve each case separately and combine the solutions to obtain the final answer.
For example, to solve the inequality $|x - 3| \leq 5$:
Case 1: $x - 3 \geq 0$, so $x \geq 3 - 5 = -2$
Case 2: $x - 3 < 0$, so $x < 3 + 5 = 8$
Combining the two cases, the solution is $-2 \leq x < 8$.
Describe the relationship between absolute value inequalities and linear inequalities, and explain how they can be used together to solve more complex problems.
Absolute value inequalities and linear inequalities are closely related, as absolute value inequalities can be rewritten as a system of linear inequalities. When solving an absolute value inequality, you essentially need to consider two linear inequalities: one where the expression inside the absolute value is greater than or equal to 0, and one where the expression is less than 0.
By understanding the connection between absolute value inequalities and linear inequalities, you can use them together to solve more complex problems. For example, you might encounter a situation where you need to find the range of values that satisfy both an absolute value inequality and a linear inequality. In such cases, you would need to solve the two inequalities simultaneously and find the intersection of their solution sets.
Analyze the role of absolute value inequalities in modeling real-world situations, and discuss how the solutions to these inequalities can be interpreted in the context of the problem.
Absolute value inequalities are often used to model real-world situations where there is a need to describe a range of acceptable values or a distance from a target value. For example, in quality control, an absolute value inequality could be used to specify the acceptable range of measurements for a product, ensuring that the measurements fall within a certain distance from the desired value. In finance, absolute value inequalities could be used to describe the acceptable range of fluctuations in stock prices or interest rates.
By analyzing the solutions to absolute value inequalities in the context of the real-world problem, you can gain valuable insights and make informed decisions. For instance, the solution set represented by the absolute value inequality might indicate the range of values that meet certain criteria or the distance from a target value that is considered acceptable. This information can be used to make informed decisions, set appropriate thresholds, or identify areas for improvement.
A linear inequality is an inequality that can be written in the form $ax + b \geq c$, $ax + b > c$, $ax + b \leq c$, or $ax + b < c$, where $a$, $b$, and $c$ are real numbers and $x$ is a variable.