Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the solution sets of the given inequalities. $$ \left|\frac{x}{4}+1\right|<1 $$

Step-by-Step Solution

Answer

The solution set is \(-8 < x < 0\).

Step 1: Understand the Absolute Value Expression

The expression inside the absolute value is \( \frac{x}{4} + 1 \). Since we are dealing with an inequality involving absolute value, the expression \( |\frac{x}{4} + 1| < 1 \) means that the expression inside the absolute value should be between -1 and 1.

Step 2: Set Up the Compound Inequality

The inequality \( |\frac{x}{4} + 1| < 1 \) can be rewritten as the compound inequality: \[-1 < \frac{x}{4} + 1 < 1.\]

Step 3: Solve the Left Inequality

Let's tackle the left part of the compound inequality first: \[-1 < \frac{x}{4} + 1.\] Subtract 1 from both sides:\[-2 < \frac{x}{4}.\] Then multiply both sides by 4 to solve for \(x\): \[x > -8.\]

Step 4: Solve the Right Inequality

Now solve the right part of the compound inequality:\[\frac{x}{4} + 1 < 1.\] Subtract 1 from both sides:\[\frac{x}{4} < 0.\] Then multiply both sides by 4 to solve for \(x\): \[x < 0.\]

Step 5: Combine the Solutions

From Steps 3 and 4, we have:\(x > -8\) and \(x < 0\).Thus, the solution set for the inequality \( |\frac{x}{4} + 1| < 1 \) is the intersection of these two inequalities:\(-8 < x < 0.\)

Key Concepts

Absolute Value Inequality

In mathematics, an absolute value inequality involves expressions with an absolute value symbol. Absolute value, denoted by vertical lines such as \(|a|\), represents the distance of a number from zero on the number line, eliminating any negative sign. For instance, \(|3| = 3\) and \(|-3| = 3\). When solving an absolute value inequality like \(|\frac{x}{4} + 1| < 1\), it implies that the expression inside the absolute value, \(\frac{x}{4} + 1\), should lie within a specified range.

This type of inequality can be translated into a compound inequality because the absolute value specifies that the expression must be both greater than the negative of the number and less than the number itself. In other words, \(|A| < B\) converts to \(-B < A < B\). Understanding this conversion can substantially simplify solving absolute value inequalities in practical scenarios.

Compound Inequality

A compound inequality consists of two separate inequalities that are linked by the words 'and' or 'or.' In our exercise, \(-1 < \frac{x}{4} + 1 < 1\) is an example of a compound inequality formed from the absolute value inequality.

This new statement means we have two conditions to satisfy simultaneously: \(-1 < \frac{x}{4} + 1\) and \(\frac{x}{4} + 1 < 1\). The 'and' indicates both conditions must be true at the same time for the solution to be valid.

Solving a compound inequality involves addressing each inequality separately. We improve our understanding by recognizing that compound inequalities help us identify a range of solutions that make the initial inequality true.

• Solve each part as a singular inequality
• Find the overlap or common solution

This will lead us directly to the solution set.

Solution Set

A solution set is the collection of all values that satisfy a given inequality or equation. In this problem, the solution set for the inequality \(|\frac{x}{4} + 1| < 1\) corresponds to the overlapping values of \(x\) that make the compound inequality \(-1 < \frac{x}{4} + 1 < 1\) true.

After analyzing each side of the inequality, we find that \(x > -8\) and \(x < 0\). Consequently, the solution set is the intersection of these two results, expressed as \(-8 < x < 0\). This means any number between -8 and 0, excluding -8 and 0 themselves, will satisfy the original inequality.

• Represent the solution set properly
• Check that each part of the original inequality holds within the found range

The solution set simplifies our understanding of the problem and provides insight into the values of \(x\) that work.

Mathematical Problem Solving

Mathematical problem solving often involves converting complex expressions into simpler forms to find solutions. In this context, understanding absolute value inequalities and their conversion into compound inequalities helps streamline the process.

Key steps include:

• Interpreting the problem and breaking down expressions.
• Converting absolute value inequalities to compound inequalities.
• Solving each inequality separately.
• Identifying the intersection of the solutions to form the solution set.

Insights gained from mathematical problem solving are invaluable; they enhance our ability to tackle similar issues with confidence. This exercise serves as a reminder of the importance of methodical problem-solving strategies in mathematics. Keeping strategies clear and straightforward ensures that even complex problems become manageable. Through consistent practice, these strategies become second nature, paving the way for easier mastery of more advanced mathematical concepts.