Problem 18 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ -\frac{1}{4}(2 z-6)+z \geq 5 $$

Step-by-Step Solution

Answer

The solution is \([7, \infty)\).

Step 1: Distribute the Fraction

Start by distributing \(-\frac{1}{4}\) to \(2z - 6\). This gives us \(-\frac{1}{4} \cdot 2z + \frac{1}{4} \cdot 6 = -\frac{1}{2}z + \frac{3}{2}\).

Step 2: Simplify the Expression

The inequality is now: \(-\frac{1}{2}z + \frac{3}{2} + z \geq 5\). Simplify by combining like terms: \(z - \frac{1}{2}z = \frac{1}{2}z\). Thus, the inequality becomes \(\frac{1}{2}z + \frac{3}{2} \geq 5\).

Step 3: Isolate the Variable Term

First, eliminate the constant term from the inequality by subtracting \(\frac{3}{2}\) from both sides: \(\frac{1}{2}z + \frac{3}{2} - \frac{3}{2} \geq 5 - \frac{3}{2}\), which simplifies to \(\frac{1}{2}z \geq \frac{10}{2} - \frac{3}{2} = \frac{7}{2}\).

Step 4: Solve for the Variable

Multiply both sides by 2 to isolate \(z\): \(2 \cdot \frac{1}{2}z \geq 2 \cdot \frac{7}{2}\), giving \(z \geq 7\).

Step 5: Express the Solution

Since the inequality \(z \geq 7\) means that \(z\) is equal to or greater than 7, express the solution in interval notation as \([7, \infty)\).

Key Concepts

Set-Builder Notation

Set-builder notation is a way to describe a set by specifying a property that its members must satisfy. In our inequality problem, once we solved for \(z\), we found that it must be greater than or equal to 7. Using set-builder notation, this solution can be expressed as:

• \(\{ z \mid z \geq 7 \}\), which reads as: "the set of all \(z\) such that \(z\) is greater than or equal to 7".

This notation is useful because it provides a clear description of the solution set based on the condition. We present the solutions in terms of the condition that each element must satisfy to be a part of the set.
Set-builder notation is particularly helpful in mathematics because it provides a concise and precise way of expressing the set of solutions for inequalities and other algebraic problems.

Interval Notation

Interval notation is an alternative to set-builder notation that uses parentheses or brackets to describe the solutions of an inequality. In our solution, we found that \(z \geq 7\). This translates to the interval notation:

• \([7, \infty)\)

The bracket \([\) next to the 7 indicates that 7 is included in the set (closed interval), while the parenthesis \(()\) next to infinity \(\infty\) signifies that it extends indefinitely without including infinity (open interval).
This notation is often preferred in calculus and higher mathematics due to its simplicity and clarity in representing boundless quantities like infinity. It also visually straightforward as it translates the condition directly into on the number line.

Distributive Property

The distributive property is a fundamental property in algebra that involves distributing one number over addition or subtraction within parentheses. In solving our inequality, the first step was to apply the distributive property:

• \(-\frac{1}{4}(2z - 6) = -\frac{1}{4} \cdot 2z + \frac{1}{4} \cdot 6\)
• This simplifies to \(-\frac{1}{2}z + \frac{3}{2}\)

The distributive property allows us to break down expressions involving multiplication into simpler parts, which can then be combined or solved more easily.
It's essential to maintain accuracy during distribution, especially with negative numbers.
The property can be best understood through practice, as it helps establish a basic skill for manipulating algebraic expressions efficiently.