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Magazine Chapter 0: Problem 36
Find the solution sets of the given inequalities. $$ |x+2|<1 $$
Short Answer
Expert verified
The solution set is \((-3, -1)\).
Step by step solution
01
Understanding the Absolute Value Inequality
The inequality given is \(|x+2| < 1\). Absolute value inequalities express the distance of a number from zero on the number line. Here, \(|x + 2| < 1\) implies that the distance between \(x + 2\) and zero is less than 1.
02
Rewriting the Inequality
The property of absolute values states that if \(|a| < b\), then \(-b < a < b\). Applying this to \(|x+2| < 1\), we have: \[-1 < x+2 < 1\].
03
Solving the Double Inequality
Now solve the double inequality \(-1 < x+2 < 1\):1. Subtract 2 from all parts of the inequality: \[-1 - 2 < x + 2 - 2 < 1 - 2\]2. Simplify the inequality: \[-3 < x < -1\].
04
Interpreting the Solution Set
The solution \(-3 < x < -1\) means that \(x\) can be any number between \(-3\) and \(-1\), not inclusive of \(-3\) or \(-1\). The solution set can be expressed in interval notation as \((-3, -1)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inequality Solving Steps
Inequalities can seem a bit tricky, but with a clear step-by-step approach, they become much easier to handle. Let's break down how we solve absolute value inequalities like \(|x+2| < 1\).
- Step 1: Understand the Absolute Value
Absolute value represents the distance from zero, so \(|x+2| < 1\) tells us that \(x+2\) is less than 1 unit away from zero. - Step 2: Remove the Absolute Value
We convert the absolute value inequality \(|x+2| < 1\) into a double inequality: \(-1 < x+2 < 1\). This helps us examine the impurity in simpler terms. - Step 3: Solve the Double Inequality
This involves manipulating each part of the inequality. Subtract 2 from every section, resulting in \-3 < x < -1\. - Step 4: Interpret the Solution
Read the solution as the range of values \(x\) can take. In this case, \(-3 < x < -1\) implies any number \(x\) between -3 and -1 without including these endpoints.
Double Inequality
A double inequality is like a sandwich, where the unknown variable is squeezed between two boundary numbers. In simple terms, it's a way of saying that one thing is less than another, which is less than a third thing. For example, \(-1 < x+2 < 1\) from our exercise is a double inequality.
By following the double inequality, we're solving a compound statement. You have to treat it as two separate parts: \(-1 < x+2\) and \(x+2 < 1\). Solving both will give you the correct range for \(x\).
By following the double inequality, we're solving a compound statement. You have to treat it as two separate parts: \(-1 < x+2\) and \(x+2 < 1\). Solving both will give you the correct range for \(x\).
- Handling Left Part: Treat \(-1 < x+2\) and solve for \(x\) which begins by subtracting 2.
- Handling Right Part: Similarly, solve \(x+2 < 1\) by also subtracting 2 to keep harmony in the inequality.
Doing this across the board maintains balance and precision when handling double inequalities.
Interval Notation
Once we solve an inequality and have a range for \(x\) like \-3 < x < -1\, we need a neat way to write that interval. That's where interval notation comes into play.
Interval notation is shorthand for expressing the set of solutions easily. Instead of writing a condition like \(-3 < x < -1\), we use \((-3, -1)\) in interval notation.
This format, \((-3, -1)\), clearly states that \(x\) is greater than -3 and less than -1, excluding both -3 and -1.
Interval notation is shorthand for expressing the set of solutions easily. Instead of writing a condition like \(-3 < x < -1\), we use \((-3, -1)\) in interval notation.
This format, \((-3, -1)\), clearly states that \(x\) is greater than -3 and less than -1, excluding both -3 and -1.
- Parentheses: Indicate that the boundary numbers aren't included in the set, which aligns with "less than" and "greater than" conditions in inequality.
- Brackets: In other contexts, using brackets like \[a, b\] means including boundaries, as with \("less than or equal to"\).
