Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand Absolute Inequality

We are given the inequality \(|x+2| < 1\). This means that the distance from \(x+2\) to 0 is less than 1 on the number line. In simple terms, \(x+2\) lies between -1 and 1.

Step 2: Rewrite the Compound Inequality

The inequality \(|x+2| < 1\) can be rewritten as the compound inequality: \[-1 < x+2 < 1.\] We are able to rewrite it this way because the absolute value function represents distance, meaning the expression inside the absolute value must lie between the negative and positive of the given number.

Step 3: Solve for x

Next, we solve the compound inequality for \(x\). Start by isolating \(x\): 1. From \(-1 < x+2\), subtract 2 from all sides: \[-1 - 2 < x < 1 - 2,\] which simplifies to \(-3 < x < -1.\)Thus, the solution set in interval notation for \(x\) is \((-3, -1)\).

Key Concepts

Compound Inequalities

In mathematics, compound inequalities involve two distinct inequalities joined by the words "and" or "or". They help define the range of values that satisfy a condition, allowing us to describe complex relationships using simpler conditions.
When dealing with an inequality like \(|x+2| < 1\), we break it down into two inequalities:

• \(-1 < x+2 < 1\)

This results in a band of solutions where both conditions must be true simultaneously. If either side of the compound inequality is not met, the solution is invalid.
The term "compound" signifies a combination, in this case, combining two linear inequalities. This makes it easier to understand and solve problems where a variable must satisfy several conditions at once. Remember, solving compound inequalities usually involves treating the two inequalities separately at first and later combining their results.

Interval Notation

Interval notation is a shorthand that expresses the set of solutions for inequalities, helping to communicate solutions clearly and concisely.
It uses parentheses and brackets to describe intervals of values:

• Parentheses \(( )\) indicate values are not included (open interval).
• Brackets \([ ]\) denote that values are included (closed interval).

For the compound inequality \(-3 < x < -1\), the solution is written in interval notation as \((-3, -1)\). It means that \(x\) can take any value greater than \(-3\) but less than \(-1\), but \(-3\) and \(-1\) themselves are not part of the solution.
Interval notation is useful because it provides a clear and standardized way to express solutions, making it easier to communicate with others and ensuring that solution sets are easily understood.

Solving Inequalities

Solving inequalities is similar to solving equations, but with additional rules to consider, especially when involving absolute values and compound forms.
When solving \(|x+2| < 1\) for \(x\), it is key to think about the absolute value as a distance from zero.
To get rid of the absolute value, rewrite it as a compound inequality:

• Divide the inequality into \(-1 < x+2 < 1\).

Next, solve each part independently by subtracting 2 from all sides:

• From \(-1 < x+2\), get \(-3 < x\).
• From \(x+2 < 1\), get \(x < -1\).

Combine these results to form \(-3 < x < -1\).
This results in \(x\) falling between \(-3\) and \(-1\). Always remember that when solving inequalities, the direction of the inequality sign matters, especially when multiplying or dividing by negative numbers—though that's not applied here, it's a crucial tip for inequality solutions in general.