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 \((-8, 0)\).

Step 1: Understand the Absolute Value Inequality

When dealing with an absolute value inequality, such as \( |A| < B \), it implies that \(-B < A < B\). In this problem, we have \( \left| \frac{x}{4} + 1 \right| < 1 \).

Step 2: Break Down the Inequality

Apply the definition from Step 1 to write two separate inequalities:\[-1 < \frac{x}{4} + 1 < 1\].

Step 3: Solve the Left Inequality

Solve the inequality \(-1 < \frac{x}{4} + 1\) by isolating \(x\):Subtract 1 from both sides: \(-2 < \frac{x}{4}\).Multiply both sides by 4 to solve for \(x\): \( -8 < x\).

Step 4: Solve the Right Inequality

Solve the inequality \(\frac{x}{4} + 1 < 1\) by isolating \(x\):Subtract 1 from both sides: \(\frac{x}{4} < 0\).Multiply both sides by 4 to solve for \(x\): \( x < 0\).

Step 5: Combine the Solution Sets

Combine the results of Step 3 and Step 4 to find the solution set.The solution to the inequality is \(-8 < x < 0\), or in interval notation, \((-8, 0)\).

Key Concepts

Absolute Value

Absolute value is a concept that refers to the distance of a number from zero on the number line, irrespective of direction. This is something central to various types of inequalities. When dealing with an inequality that includes an absolute value, we use the idea that

• anything contained within the absolute value can have both a positive and a negative value, as long as it does not exceed the given boundaries.
• For example, in the inequality \(|A| < B\), it implies \(-B < A < B\), showcasing the range of values the expression can take.

In the problem at hand, we manage the absolute value by breaking it into two separate inequalities. This helps visualize potential values for \(A\). Understanding absolute value helps us factor different scenarios into a set of equations.

Solution Sets

The term "solution set" describes all possible values that a variable can take to satisfy an equation or inequality completely.

Once you express the absolute value inequality as two separate inequalities, you solve each to identify permissible values of the variable.

In this specific exercise, one inequality gives us \(-8 < x\), and another produces \(x < 0\).

Combining these results provides the entire range of values meeting the original condition.

• A solution set expresses this relationship and captures all valid solutions.
• This range can be considered graphically drawn on a number line as a line segment between two points.

Grasping this idea ensures accuracy when conveying multiple possibilities in problem-solving within mathematical constraints.

Interval Notation

Interval notation is a concise method to describe the range of values that a variable can assume within the solution set of an inequality.

Using interval notation helps in avoiding any misinterpretation of with which values the variable is associated.

For example, the inequality solution \(-8 < x < 0\) is translated into interval notation as \((-8, 0)\).

• The parentheses \(()\) indicate that neither \(-8\) nor \(0\) is included in the solution set.
• If the endpoints were included, we would use brackets \([\ ]\).
• This method simplifies writing the solution and is universally understood for defining a range of solutions.

Interval notation is especially valuable in representing continuous ranges or segments of a number line, making it an important tool in mathematical expressions.