Problem 17 - College Algebra

Question

Solve the inequality involving absolute value. Write your final answer in interval notation. $$ |3 x-1|>11 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -\frac{10}{3}) \cup (4, \infty)\).

Step 1: Understanding the Absolute Value Inequality

We begin by recalling that the absolute value inequality \(|A| > B\) translates to two inequalities: \(A > B\) or \(A < -B\). So, for the inequality \(|3x - 1| > 11\), we set up the following two inequalities:1. \(3x - 1 > 11\)2. \(3x - 1 < -11\)

Step 2: Solving the First Inequality

First, solve \(3x - 1 > 11\):- Add 1 to both sides to isolate the term with \(x\): \[ 3x > 12 \]- Divide both sides by 3 to solve for \(x\): \[ x > 4 \]

Step 3: Solving the Second Inequality

Next, solve \(3x - 1 < -11\):- Add 1 to both sides: \[ 3x < -10 \]- Divide both sides by 3 to solve for \(x\): \[ x < -\frac{10}{3} \]

Step 4: Writing the Solution in Interval Notation

The solutions from both inequalities are combined to get the final answer in interval notation. The solution is the union of the intervals from both inequalities: \(x > 4\) or \(x < -\frac{10}{3}\). In interval notation, this is written as:\[ (-\infty, -\frac{10}{3}) \cup (4, \infty) \]

Key Concepts

Interval Notation

Interval notation is a concise way of expressing a range of values, often used in mathematics to describe the set of solutions for inequalities. It's an essential tool because it helps represent complicated solution sets in a simple and readable manner.
Using interval notation involves certain symbols:

• Square brackets [ ] indicate that an endpoint is included (called a closed interval).
• Parentheses ( ) indicate that an endpoint is not included (called an open interval).
• Infinity (∞) is always expressed with a parenthesis because infinity is a concept and cannot be "included" in the traditional sense.

When we solved the inequality \(|3x - 1| > 11\), our solution consisted of two separate intervals. We used the union symbol (∪) to combine them, which reads in interval notation as: \[ (-\infty, -\frac{10}{3}) \cup (4, \infty) \]This notation tells us that the solution includes all numbers less than \(-\frac{10}{3}\) and all numbers greater than 4.

Solving Inequalities

Solving inequalities is about finding the range of values that satisfy a given inequality. Inequalities can signify that one expression is greater than, less than, or even equal to another. Solving inequalities involving absolute values requires understanding and breaking down the expressions into manageable parts.
For the absolute value inequality \(|3x - 1| > 11\), we start by transforming it into two separate inequalities:

• \(3x - 1 > 11\)
• \(3x - 1 < -11\)

These two inequalities derive from the definition of absolute value, which considers both the positive and negative scenarios of the expression inside. Once the inequalities are established, the process is straightforward:

• Add or subtract terms to isolate the variable terms.
• Divide or multiply to solve for the variable while ensuring to flip the sign if multiplying or dividing by a negative number.

Successfully solving these gives the set of conditions where the original inequality is true.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and include numbers, variables, and arithmetic operations. Understanding how to manipulate these expressions is crucial in solving equations and inequalities.
In the inequality we've dealt with, \(3x - 1\), we see a linear expression. When solving inequalities, our primary goal is to isolate the variable on one side of the inequality. We do this by performing operations like addition, subtraction, multiplication, or division:

• For \(3x - 1 > 11\), adding 1 to both sides first gives \(3x > 12\).
• Then, dividing by 3 isolates \(x\) to provide \(x > 4\).
• Similarly, solving \(3x - 1 < -11\) by adding 1 gives \(3x < -10\), and dividing by 3 provides \(x < -\frac{10}{3}\).

These steps involve transforming the algebraic expression by maintaining equality or inequality to ensure the solution accurately reflects the conditions described in the original problem.