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Magazine Chapter 1: Problem 73
Solve the absolute value inequality. Express the answer using interval notation and graph the solution set. $$|x+6|<0.001$$
Short Answer
Expert verified
The solution is \((-6.001, -5.999)\).
Step by step solution
01
Understanding Absolute Value Inequality
The given inequality is \(|x + 6| < 0.001\). The absolute value \(|a| < b\) indicates the expression inside the absolute value lies between \(-b\) and \(b\). Accordingly, we need to solve the inequality \(-0.001 < x + 6 < 0.001\).
02
Solve Lower Bound
The first part of the inequality is \(-0.001 < x + 6\). Subtract 6 from both sides to isolate \(x\): \(-0.001 - 6 < x\) This simplifies to \(-6.001 < x\).
03
Solve Upper Bound
The second part of the inequality is \(x + 6 < 0.001\). Again, subtract 6 from both sides: \(x < 0.001 - 6\) This simplifies to \(x < -5.999\).
04
Combine Solutions
Combining the two inequalities, we have \(-6.001 < x < -5.999\). This inequality shows that \(x\) can be any number between \(-6.001\) and \(-5.999\).
05
Express in Interval Notation
In interval notation, the solution is expressed as \((-6.001, -5.999)\). This includes all values between \(-6.001\) and \(-5.999\), but not the endpoints.
06
Graph the Solution Set
Graphically, plot this on a number line by drawing an open interval between \(-6.001\) and \(-5.999\). Because the inequality is strict \((<)\), use open circles to indicate that the endpoints are not included in the solution set.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a neat and compact way to express a range of numbers in a mathematical solution. Instead of writing out all the numbers in an inequality, we represent the entire set of numbers in a simple way.
For example, the solution to \(|x + 6| < 0.001\) is written as \((-6.001, -5.999)\) using interval notation. Here's how it works:
This notation allows you to quickly see the range of solutions without writing a lengthy inequality. It provides a concise visual representation of all the possible solutions.
For example, the solution to \(|x + 6| < 0.001\) is written as \((-6.001, -5.999)\) using interval notation. Here's how it works:
- The numbers at the ends, \-6.001\ and \-5.999\, define the bounds of the interval.
- Parentheses \(()\) are used instead of brackets \([]\) because the inequality is strict \(<\), indicating that the endpoints are not included in the interval.
This notation allows you to quickly see the range of solutions without writing a lengthy inequality. It provides a concise visual representation of all the possible solutions.
Graphing Inequalities
Graphing inequalities on a number line offers a visual representation of a range of solutions. It helps you understand what part of the number line satisfies the inequality. Let's break it down:
To graph the solution to the inequality \(-6.001 < x < -5.999\), you would do the following:
Graphing inequalities like this provides a clear visual insight into the range of solutions and which values of \(x\) satisfy the inequality. It's a powerful tool for checking your work and making connections between algebraic and visual representations.
To graph the solution to the inequality \(-6.001 < x < -5.999\), you would do the following:
- Draw a number line with marks indicating \-6.001\ and \-5.999\.
- Use open circles at these points since the inequality is strict and does not include the endpoints.
- Shade the portion between these open circles, illustrating that any point in this segment of the number line is a solution.
Graphing inequalities like this provides a clear visual insight into the range of solutions and which values of \(x\) satisfy the inequality. It's a powerful tool for checking your work and making connections between algebraic and visual representations.
Solving Inequalities
Solving inequalities is a critical skill in mathematics, especially when dealing with absolute values. An inequality such as the one given, \(|x + 6| < 0.001\), requires breaking it down into a pair of simpler inequalities to find all possible solutions.
Here's a step-by-step approach:
Understanding and solving inequalities is fundamental for tackling more complex equations and represents a core aspect of algebra.
Here's a step-by-step approach:
- Start with the definition of the absolute value inequality: \(|a| < b\ means \-b < a < b\).
- Apply this to the given inequality to create \-0.001 < x + 6 < 0.001\.
- Solve each part of the inequality separately to determine the range of values for \(x\). In this case:
- Solve \-0.001 < x + 6\ by subtracting 6 from both sides to get \-6.001 < x\.
- Solve \ x + 6 < 0.001\ similarly, resulting in \ x < -5.999\.
- Combine the solutions: The inequalities combine to form \-6.001 < x < -5.999\, giving a complete picture of possible values of \(x\) satisfying the original inequality.
Understanding and solving inequalities is fundamental for tackling more complex equations and represents a core aspect of algebra.
