Problem 79 - College Algebra

Question

Solve absolute value inequality. \(3|x-1|+2 \geq 8\)

Step-by-Step Solution

Answer

\(x \leq -1\) or \(x \geq 3\)

Step 1: Simplify the equation

First, subtract 2 from both sides of the equation to isolate the absolute value term. This results in \(3|x-1| \geq 6\). Then, divide both sides of the inequality by 3 to further isolate the absolute value term giving \(|x-1| \geq 2\). This is the inequality that needs to be solved.

Step 2: Resolve the absolute value

Next, resolve the absolute value. If \(y = |x-1|\) then the possible scenarios are \(y=x-1\) and \(y= -(x-1)\) . Apply these scenarios to inequalities \(|x-1| \geq 2\), which results in two inequalities \(x-1 \geq 2\) and \(-(x-1) \geq 2\).

Step 3: Solve the inequalities

Now solve each inequality separately. Adding 1 to both sides of the first inequality gives \(x \geq 3\). For the second inequality, multiply through by -1 (and reverse the inequality symbol) to give \(x-1 \leq -2\) and then, adding 1 to both sides gives \(x \leq -1\).

Step 4: Combination of the results

The solutions to the original inequality are the combination of the solutions to the two separate inequalities. Therefore, \(x\) could be anything less than or equal to -1 or greater than or equal to 3. This is the final solution to the given absolute value inequality.

Key Concepts

Solving Inequalities

Inequalities express a relationship where two expressions may not be equal. They tell us how one expression relates to another. In math, commonly used inequality symbols include:

• \( > \) - Greater than
• \( < \) - Less than
• \( \geq \) - Greater than or equal to
• \( \leq \) - Less than or equal to

To solve inequalities, the process is similar to solving equations. However, keep in mind that when you multiply or divide both sides by a negative number, the inequality sign flips. This is key when working through problems involving inequalities. In the absolute value inequality \(3|x-1|+2 \geq 8\), you treat both sides like an equation to isolate the absolute value expression. Once isolated, it's easier to resolve specific scenarios that satisfy the inequality. This involves addressing both positive and negative possibilities.

Algebraic Expressions

Algebraic expressions are powerful tools in mathematics. They consist of numbers, variables (like \(x\)), and arithmetic operations. They are used extensively to represent real-world scenarios in mathematical form.
For instance, in the inequality \(3|x-1| \geq 6\), \(3|x-1|\) is an algebraic expression where three multiplies the absolute value of \(x-1\). This expression can be rewritten and manipulated to simplify the inequality.
Understanding the structure of these expressions allows us to rearrange the terms using operations like addition, subtraction, multiplication, and division. In problem-solving, identify the like terms to combine and isolate the necessary terms to further simplify the challenge. This process helps in deriving the solution step by step. As seen, algebraic manipulations helped us bring the problem to determine conditions for \( x \), so it's less daunting.

Absolute Value Properties

Absolute value represents the distance of a number from zero on the number line, without considering direction. This means \(|x|\) is always positive or zero.
An absolute value inequality like \(|x-1| \geq 2\) sets a condition where the distance between \(x\) and 1 is at least 2. To solve it, you translate it into a range of numbers that satisfy the condition. This involves breaking the problem into two parts:

• When the expression inside the absolute value is non-negative - solve \(x - 1 \geq 2\)
• When the expression inside the absolute value is negative - solve \(-(x - 1) \geq 2\) which simplifies to \(x - 1 \leq -2\)

These split scenarios help you cover all possibilities, ensuring you do not miss any potential solution. Thus, understanding absolute value properties simplifies solving a variety of inequality problems.