Problem 76 - College Algebra

Question

Solve absolute value inequality. \(\left|\frac{3 x-3}{9}\right| \geq 1\)

Step-by-Step Solution

Answer

The solution to the inequality is \( x \geq 4 \) or \( x \leq -2 \).

Step 1: Solve \( \frac{3x - 3}{9} \geq 1 \)

This implies \( 3x - 3 \geq 9 \), which simplifies to \( 3x \geq 12 \) and finally to \( x \geq 4 \).

Step 2: Solve \( \frac{3x - 3}{9} \leq -1 \)

This implies \( 3x - 3 \leq -9 \), which simplifies to \( 3x \leq -6 \) and finally to \( x \leq -2 \).

Step 3: Combine the results

The solutions to the two previous steps are combined into the final result: \( x \geq 4 \) or \( x \leq -2 \).

Key Concepts

Understanding Inequalities and Their Role in Solving Problems

When solving inequality problems, especially those involving absolute values, we often deal with expressions that define ranges of values a variable can take. Absolute value inequalities like \( \left| \frac{3x-3}{9} \right| \geq 1 \) ask us to find all possible values for \( x \) that satisfy the inequality. This means our solution will not be a single number but a set of numbers, often expressed as a range.

The first step when tackling absolute value inequalities is to understand that the expression within the absolute value can have both positive and negative scenarios. Inequalities come in two general forms:

• \( \frac{3x - 3}{9} \geq 1 \)
• \( \frac{3x - 3}{9} \leq -1 \)

By solving each of these inequalities separately, we determine the ranges for \( x \). Remember, when dealing with absolute values, we consider the expression inside the absolute value as if it can be both greater than or equal to and less than or equal to a specific number.

Manipulating Algebraic Expressions for Inequalities

Algebraic expression manipulation is all about rearranging the terms of an equation to isolate the variable. It's much like solving a puzzle. For absolute value inequalities, this often involves breaking the problem into two separate inequalities. In this case, those inequalities are \( \frac{3x-3}{9} \geq 1 \) and \( \frac{3x-3}{9} \leq -1 \).

To solve these effectively:

• First, multiply both sides by 9 to eliminate the fraction: \( 3x - 3 \geq 9 \), and \( 3x - 3 \leq -9 \).
• Next, isolate the term with \( x \) by adding or subtracting any constants: \( 3x \geq 12 \) and \( 3x \leq -6 \).
• Finally, divide by the coefficient of \( x \), which is 3, to solve for \( x \): \( x \geq 4 \) and \( x \leq -2 \).

Each step in the manipulation helps simplify the inequality so that the solution becomes clearer and easier to interpret.

Step-by-Step Process to Solve the Given Inequality

Mathematical solution steps are a systematic way to tackle problems. Here, the process involves a few key operations that unfold sequentially. Let's walk through the process used in our exercise to ensure clarity:

**Step 1:** Solve \( \frac{3x - 3}{9} \geq 1 \). Multiply by 9, then move all constant terms to one side, resulting in \( x \geq 4 \).

**Step 2:** Solve \( \frac{3x - 3}{9} \leq -1 \). Similarly, multiply by 9 to clear the fraction, then shift the unwanted terms to arrive at \( x \leq -2 \).

**Step 3:** Combine the results from both inequalities, summarizing the solution to be \( x \geq 4 \) or \( x \leq -2 \). Notice that these steps allow you to evaluate the behavior of \( x \) in different ranges. This sequential breakdown transforms a potentially overwhelming problem into manageable parts. By focusing on each step, you build a comprehensive understanding of the inequality.