Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation or inequality. $$|-3 x+8| \geq 3$$

Step-by-Step Solution

Answer

The solution is \(x \leq \frac{5}{3}\) or \(x \geq \frac{11}{3}\).

Step 1: Understand Absolute Value Inequality

The expression \(|-3x + 8| \geq 3\) means the distance of \(-3x + 8\) from zero is at least 3. This implies two cases: 1. \(-3x + 8 \geq 3\)2. \(-3x + 8 \leq -3\)

Step 2: Solve Case 1

For the inequality \(-3x + 8 \geq 3\), start by subtracting 8 from both sides:\(-3x + 8 - 8 \geq 3 - 8\)This simplifies to:\(-3x \geq -5\)Divide both sides by \(-3\) and remember to flip the inequality sign:\(x \leq \frac{5}{3}\)

Step 3: Solve Case 2

For the inequality \(-3x + 8 \leq -3\), start by subtracting 8 from both sides:\(-3x + 8 - 8 \leq -3 - 8\)This simplifies to:\(-3x \leq -11\)Divide both sides by \(-3\) and remember to flip the inequality sign:\(x \geq \frac{11}{3}\)

Step 4: Combine Solutions

For the original absolute value inequality \(|-3x + 8| \geq 3\), the solutions are valid from the combination of the two individual inequalities. Thus, either:1. \(x \leq \frac{5}{3}\), or2. \(x \geq \frac{11}{3}\)This is the union of two regions.

Key Concepts

Solving Inequalities

When solving inequalities, especially absolute value inequalities like \(|-3x + 8| \geq 3\), the key is breaking down the problem into manageable parts. Absolute value inequalities express a range of distances from zero, which leads to creating two separate conditions to solve. Let's explore how this unfolds: - Firstly, convert the absolute value statement into two inequalities based on the critical principle of distance: 1. \(-3x + 8 \geq 3\) and 2. \(-3x + 8 \leq -3\). - Why two conditions? The absolute value represents the magnitude or distance without considering direction, hence both cases naturally arise.
Simplifying these inequalities involves algebraic manipulations. Solve each case independently:

• Subtract 8 from both sides to keep the expression balanced.
• Divide by \(-3\), flipping the inequality sign to maintain the direction of inequality.

These steps unravel the solution sets that reveal where the expression satisfies the given condition.

Inequality Solutions

The solutions to inequalities offer insight into the range of values that fulfill the given condition. For the inequality \(|-3x + 8| \geq 3\), solving the two conditions separately yields the solutions: \(x \leq \frac{5}{3}\) and \(x \geq \frac{11}{3}\).Let's break it down: - 1. Solving \(-3x + 8 \geq 3\) gives \(x \leq \frac{5}{3}\).2. Solving \(-3x + 8 \leq -3\) results in \(x \geq \frac{11}{3}\). - These are not solutions for a single interval. Instead, they describe two separate regions that do not overlap. - Collectively, they form a union of solutions where either condition satisfies the inequality.
The broader implication of these solutions lies in their real-world relevance, whereby multiple conditions could simultaneously define limits. This teaches us that solutions can appear in various distinct ranges, illustrating how values can exist in separate spectrums depending on restrictions.

Mathematical Reasoning

Mathematical reasoning empowers us to approach problems systematically. When dealing with absolute value inequalities, reason through how two divergent inequalities stem from a single expression. Let's think about why this makes sense: - The absolute value speaks to a number's distance from zero, either being greater than or equal to the threshold in either direction. Therefore, a single absolute value inequality becomes two complementary conditions. Mathematical reasoning means:

• Understanding the underlying principles that govern behavior or change in values, like flipping the inequality sign when multiplying or dividing by a negative number.
• Reasoning through complex problems by dissecting them into straightforward components, as seen in our two inequality cases.
• Encapsulating logic in mathematical processes, turning intuitive ideas into precise calculations and expressions.

The strength of mathematical reasoning extends past mere calculation; it involves seeing the broader pictures of how math models situations. This way of thinking transforms absolutist numbers into relational concepts, vital in understanding not just where solutions lie, but why they do.