Problem 79 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ 3|x-1|+2 \geq 8 $$

Step-by-Step Solution

Answer

The solution to the inequality \(3|x-1|+2 \geq 8\) is \(x \leq -1\) or \(x \geq 3\).

Step 1: Isolate the absolute value

Subtract 2 from both sides of the inequality to isolate the absolute value on one side. This gives: \(3|x-1| \geq 6\).

Step 2: Remove the coefficient from the absolute value

Divide both sides by 3 to remove the coefficient from the absolute value. This gives: \(|x-1| \geq 2\).

Step 3: Create two equations

Create two inequalities, turning the absolute value into a positive and a negative instance. This results in: \(x - 1 \geq 2\) and \(x - 1 \leq -2\).

Step 4: Solve the first inequality

Solving for \(x\) in the first inequality \(x - 1 \geq 2\), gives \(x \geq 3\).

Step 5: Solve the second inequality

Solving for \(x\) in the second inequality \(x - 1 \leq -2\), gives \(x \leq -1\).

Key Concepts

Solving Absolute Value Inequalities

To solve absolute value inequalities, the key is to isolate the absolute value expression before proceeding with other steps. An absolute value inequality involves expressions like \(|x-1|\), which represent the distance from zero on a number line, making it always non-negative. Here's the process:

• Isolate the Absolute Value: Remove any constants or coefficients outside the absolute value first. In our given problem \(3|x-1| + 2 \geq 8\), subtract 2 to begin isolating the absolute value, resulting in \(3|x-1| \geq 6\).
• Divide by the Coefficient: If there's a coefficient next to the absolute value, divide it away. Continue with \(|x-1| \geq 2\).

Isolating the absolute value makes transforming the inequality into simpler cases manageable. This step-by-step approach helps ensure that you're correctly setting up the expressions for additional solving techniques.

Understanding and Manipulating Algebraic Expressions

Algebraic expressions are key in setting up the inequalities for solving. These expressions, like \(x-1\), need manipulation so that you can properly set up for solving inequalities:

• Breaking into Two Cases: When dealing with expressions inside an absolute value, like \(|x-1| \geq 2\), you split it into two separate inequalities. This uses the definition that absolute values measure distance from zero, thus requiring a positive and negative scenario.
• Setting Up Equations: Thus, solving \(|x-1| \geq 2\) requires setting \(x-1 \geq 2\) and \(x-1 \leq -2\). This makes sure you capture the full range of possible solutions for \(x\).

By understanding how to rewrite these expressions into simpler formats, you make solving these inequalities more intuitive and straightforward.

Finding the Inequality Solutions

Once the absolute value is isolated and the corresponding algebraic expressions created, you proceed to find solutions to each inequality. Solving the inequalities offers clarity towards finding the range within which the variable fits:

• Solving Each Inequality: Take each separated inequality and solve individually. For \(x-1 \geq 2\), add 1 on both sides to get \(x \geq 3\). Similarly, solve \(x-1 \leq -2\), leading to \(x \leq -1\).
• Analyzing Results: The solutions \(x \geq 3\) and \(x \leq -1\) can then be combined to form the complete solution set for the original inequality. However, notice that no numbers can be simultaneously greater than or equal to 3 and less than or equal to -1, indicating a possible range or non-existent solution depending on your context.

Understanding the full picture of these solutions allows students to comprehend how absolute values and inequalities interact, further enhancing their algebraic problem-solving skills.