Question

Find the solution sets of the given inequalities. $$ |x+2|<1 $$

Short answer

The solution set is \((-3, -1)\).

Step-by-step solution

Understand the Absolute Value Inequality

The inequality \(|x+2| < 1\) is stating that the distance between \(x+2\) and 0 on a number line is less than 1. This means that \(x+2\) must lie between -1 and 1.

Break Down the Absolute Value

Since the absolute value of a number is less than 1, we can write the inequality without the absolute value as two separate inequalities: \[ -1 < x+2 < 1 \] This comes from splitting the less than inequality into two parts.

Solve the Two Inequalities Separately

We now solve each part of the compound inequality: For \(-1 < x+2\), subtract 2 from both sides to get: \[ x > -3 \] For \(x+2 < 1\), subtract 2 from both sides to get: \[ x < -1 \]

Find the Intersection

Combine the solutions from the two inequalities. The solution is where the two conditions overlap: \[-3 < x < -1\] This is the set of all \(x\) values that satisfy both inequalities.

Write the Solution Set

We express the solution set as an interval: \((-3, -1)\) This means \(x\) can be any number between -3 and -1, not inclusive of -3 and -1.

Key concepts

Compound Inequality

A compound inequality consists of two separate inequalities that are combined into one statement, often using the words "and" or "or." In the exercise, the absolute value inequality \(|x+2| < 1\) is equivalent to saying that the expression on the number line falls within a specified range. This specific exercise breaks down into two parts:\

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• \(-1 < x+2\)
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• \(x+2 < 1\)
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\Because both conditions need to be true, we join them with "and." This means we look for values of \(x\) that satisfy both inequalities together. This is a crucial aspect of solving compound inequalities, as it helps narrow down the range of possible solutions.

Solution Set

The solution set is the collection of all possible values that \(x\) can take to satisfy an inequality or system of inequalities. After setting up and solving the compound inequality, we determine the solution set.\
\For our problem, after solving each inequality, we find that \(x > -3\) and \(x < -1\). The solution set needs to fulfill both these conditions. Therefore, the combined solution set, \(-3 < x < -1\), represents all \(x\) values that lie between these two points on the number line.\
\In essence, the solution set captures all the valid numbers that satisfy the original absolute value inequality by fulfilling both compound inequality parts.

Interval Notation

Interval notation provides a concise way of expressing the solution set of inequalities. It uses parentheses and brackets to describe sets of numbers and whether endpoints are included or not.\
\In the exercise, the solution \(-3 < x < -1\) is expressed in interval notation as \((-3, -1)\), indicating all numbers between -3 and -1, where neither -3 nor -1 are included in the set.\

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• Parentheses \(()\) signify that the endpoints themselves are not part of the solution set (open interval).
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• Square brackets \([]\) would mean the endpoints are included (closed interval). In this case, the problem specifies an open interval and uses parentheses.
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\This method offers a clear and efficient way to present the range of numbers fulfilling the inequality.

Inequality Solving Steps

Solving inequalities involves a methodical approach to ensure all conditions are met. The solution process for the exercise is done in clear steps: \(\vert{x + 2} \vert < 1 \).\

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• **Understand the Absolute Value:** Recognize the inequality represents a range of values \(x + 2\) is from -1 to 1.
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• **Break Down the Absolute Value:** Split the absolute value into a compound inequality: \(-1 < x+2 < 1\).
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• **Solve Each Inequality:** Independently solve each part. From \(-1 < x+2\), subtract 2 to get \(x > -3\), and from \(x+2 < 1\), subtract 2 to get \(x < -1\).
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• **Determine the Intersection:** Find the overlap between \(x > -3\) and \(x < -1\), resulting in \(-3 < x < -1\).
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• **Express the Solution:** Use interval notation \((-3, -1)\) to present the solution set neatly.
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\These steps ensure a thorough understanding and solution of absolute value inequalities, capturing all possible valid solutions for the given problem.