Problem 25 - 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. $$ \left|\frac{z}{5}-1\right| \leq 1 $$

Step-by-Step Solution

Answer

The solution set is \([0, 10]\).

Step 1: Interpret the Absolute Inequality

Given the inequality \( \left|\frac{z}{5} - 1\right| \leq 1 \), we need to interpret it as two related inequalities. Recall that the definition of absolute value inequality \( |x| \leq a \) can be rewritten as \( -a \leq x \leq a \). Applying it here gives us: \[ -1 \leq \frac{z}{5} - 1 \leq 1. \]

Step 2: Solve the Left Inequality

Solve the first inequality from Step 1: \( -1 \leq \frac{z}{5} - 1 \). Add 1 to both sides to simplify: \[ 0 \leq \frac{z}{5}. \] Then multiply both sides by 5 to isolate \( z \): \( 0 \leq z \).

Step 3: Solve the Right Inequality

Solve the second inequality from Step 1: \( \frac{z}{5} - 1 \leq 1 \). Add 1 to both sides to simplify: \[ \frac{z}{5} \leq 2. \] Then multiply both sides by 5 to isolate \( z \): \( z \leq 10 \).

Step 4: Combine the Solution

Combine the solutions from Steps 2 and 3 to express the solution to the inequality as \( 0 \leq z \leq 10 \), or equivalently, \([0, 10]\). This indicates that \( z \) must be a number between 0 and 10, inclusive.

Step 5: Represent the Solution on the Real Line

On a number line, represent the interval \([0, 10]\) by drawing a number line and shading the region from 0 to 10, including the endpoints by using closed circles at 0 and 10.

Key Concepts

Absolute Value Inequality

Absolute value inequalities like \( \left|\frac{z}{5} - 1\right| \leq 1 \) involve the absolute difference from zero. Absolute value measures the distance on a number line, ignoring direction, and is always non-negative.
When solving this type of inequality, we interpret it as two related inequalities because it establishes limits within which expressions must lie.
For a general inequality \( |x| \leq a \), this translates to \(-a \leq x \leq a\), encompassing both positive and negative deviations.
In our specific case, this translates to

• \(-1 \leq \frac{z}{5} - 1\)
• \(\frac{z}{5} - 1 \leq 1\)

This dual setup reveals the permissible range for the expression inside the absolute value, which we then solve separately to find the range for \(z\).
The solutions of these related inequalities give us the bounds for \(z\), guiding us to form a complete picture of all possible solutions.

Interval Notation

Interval notation provides a concise way of showing solution sets, particularly when dealing with inequalities.
After solving the absolute value inequality, we represent its solution as an interval, capturing all numbers within certain bounds.
The inequality \(0 \leq z \leq 10\) is expressed as \([0, 10]\), where square brackets indicate that endpoints 0 and 10 are included (closed interval).
Interval notation is particularly useful:

• To denote continuous sets of numbers.
• To simplify representation of solution sets in algebra.

For example, if the inequality was strict, such as \(0 < z < 10\), we would use parentheses \((0, 10)\) to show exclusion of the endpoints. Interval notation is an essential skill for communicating solutions clearly and efficiently in mathematical contexts.

Real Line Representation

Visualizing solutions on the real line helps intuitively understand inequalities like those solved using absolute values.
In our case, the interval \([0, 10]\) is represented on a number line by shading the segment from 0 to 10.
Closed circles or dots at the endpoints 0 and 10 indicate these values are part of the solution.
Real line representation has benefits:

• Offers a visual way to grasp the scope of the solutions.
• Helps in checking that all parts of an inequality have been considered correctly.

Being able to switch between algebraic solutions and graphical representations builds a deeper comprehension and aids in solving more complex problems in the future.