Problem 26 - Thomas Calculus

Question

Solve the inequalities in Exercises \(19-34,\) expressing the solution sets as intervals or unions of intervals. Also, show each solution set on the real line. $$ \left|\frac{3}{2} z-1\right| \leq 2 $$

Step-by-Step Solution

Answer

The solution set is \([-\frac{2}{3}, 2]\).

Step 1: Understand the Inequality

We need to solve the inequality \( \left| \frac{3}{2}z - 1 \right| \leq 2 \). This is an absolute value inequality which represents the range of values for \( z \) that satisfies it.

Step 2: Set Up Two Inequalities

An absolute value inequality \( \left| x \right| \leq a \) implies two separate inequalities: \( -a \leq x \leq a \). So for our problem, we have: 1. \( -2 \leq \frac{3}{2}z - 1 \)2. \( \frac{3}{2}z - 1 \leq 2 \)

Step 3: Solve the First Inequality

Start with the inequality \( -2 \leq \frac{3}{2}z - 1 \). Add 1 to both sides to get:\[ -1 \leq \frac{3}{2}z \]Next, multiply both sides by \( \frac{2}{3} \) (which is the reciprocal of \( \frac{3}{2} \)):\[ -\frac{2}{3} \leq z \]

Step 4: Solve the Second Inequality

Now solve the inequality \( \frac{3}{2}z - 1 \leq 2 \). Add 1 to both sides:\[ \frac{3}{2}z \leq 3 \]Then multiply both sides by \( \frac{2}{3} \):\[ z \leq 2 \]

Step 5: Combine the Solutions

We now have two conditions: \( -\frac{2}{3} \leq z \) and \( z \leq 2 \). Combined, these inequalities mean \( -\frac{2}{3} \leq z \leq 2 \). So, the solution set in interval notation is \([-\frac{2}{3}, 2]\).

Step 6: Show the Solution on the Real Line

On the real line, plot a closed interval with filled circles at \(-\frac{2}{3}\) and \(2\) to represent the inclusive range \([-\frac{2}{3}, 2]\).

Key Concepts

Real Line Representation

When solving absolute value inequalities, such as \( \left| \frac{3}{2}z - 1 \right| \leq 2 \), it's important to visualize the solution set on the real line. A real line representation provides a straightforward visual way to see the range of values that are valid solutions. It helps in understanding which numbers satisfy the inequality and how they are distributed on the same line that we use for real numbers. To represent the solution \([ -\frac{2}{3}, 2] \) on a real line, we plot this interval:

• Locate \(-\frac{2}{3}\) and 2 on the real number line by marking points clearly.
• Draw a line or a thick segment connecting these two points to indicate that all numbers between them are included in the solution set.
• Place filled (closed) circles at the ends \(-\frac{2}{3}\) and 2 because these endpoints are part of the solution set, showing the inclusivity of the inequality.

This filled line segment represents all values \(z\) such that \(-\frac{2}{3} \leq z \leq 2\). By examining this, you get a clear understanding of the solution range.

Interval Notation

Interval notation is a concise way of writing subsets of the real number line, especially useful for expressing solutions to absolute value inequalities. It uses brackets to denote whether endpoints are included or not in the interval:

• Brackets \([\,]\) imply the endpoint is included (closed interval).
• Parentheses \((\,)\) imply the endpoint is not included (open interval).

For the inequality \(\left| \frac{3}{2}z - 1 \right| \leq 2\), the solution is derived as \(-\frac{2}{3} \leq z \leq 2\). In interval notation, this is expressed as \([ -\frac{2}{3}, 2 ]\), indicating that both endpoints \(-\frac{2}{3}\) and 2 are part of the solution. Using this notation simplifies communication of the results and helps when performing further operations like intersections and unions of intervals, often seen in problems involving multiple inequalities.

Compound Inequalities

The concept of compound inequalities arises when you deal with inequalities like \( \left| \frac{3}{2}z - 1 \right| \leq 2 \), which breaks down into two separate but related conditions. These two inequalities collectively describe a range of values:

• \( -2 \leq \frac{3}{2}z - 1 \)
• \( \frac{3}{2}z - 1 \leq 2 \)

These are solved individually, resulting in expressions that are then combined, revealing the solution as a window through which \(z\) can vary. This leads to \(-\frac{2}{3} \leq z \leq 2\), a compound inequality statement that consolidates the results. This form makes it clear what values \(z\) can take and helps visualize the solution when mapped onto the real line. Understanding compound inequalities is crucial as it aids in solving more complex problems involving multiple range conditions.