Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation. $$ |3 x-1|>11 $$

Step-by-Step Solution

Answer

\((-\infty, -\frac{10}{3}) \cup (4, \infty)\)

Step 1: Understand the Absolute Value Inequality

The absolute value inequality \(|3x - 1| > 11\) implies that the expression inside the absolute value is either greater than 11 or less than -11. We break this into two separate inequalities to solve individually.

Step 2: Solve the First Inequality

For the inequality \(3x - 1 > 11\), add 1 to both sides to get \(3x > 12\). Next, divide both sides by 3 to solve for \(x\): \(x > 4\).

Step 3: Solve the Second Inequality

For the inequality \(3x - 1 < -11\), add 1 to both sides to get \(3x < -10\). Next, divide both sides by 3 to find \(x < -\frac{10}{3}\).

Step 4: Combine the Solutions

The solutions to the inequalities are \(x > 4\) and \(x < -\frac{10}{3}\). Since these are separated by the absolute value inequality, the solution is expressed in interval notation as a union: \((-fty, -\frac{10}{3}) \cup (4, fty)\).

Key Concepts

Interval Notation

Interval notation is a way of expressing the set of solutions to an inequality by using open or closed intervals. In mathematics, it's crucial to clearly communicate the range of values that satisfy certain conditions. An open interval, denoted by parentheses like \(a, b\), means the endpoints are not included in the set. Meanwhile, a closed interval, noted with brackets like \[a, b\], includes the endpoints.

For example, if we say \(1, 5\), it means all numbers greater than 1 and less than 5. But unlike \[1, 5\], it does not include 1 and 5 themselves.

In absolute value inequalities like \(|3x - 1| > 11\), once you've solved the inequalities separately, you can express the solutions using interval notation. The solutions \(x > 4\) and \(x < -\frac{10}{3}\) are expressed as the union of two intervals, \((-\infty, -\frac{10}{3}) \cup (4, \infty)\). This notation comfortably showcases all possible values that make the inequality true, as these values lie outside the closed intervals corresponding to the solutions.

Solving Inequalities

Solving inequalities involves a process similar to solving equations, but with special attention to the direction of the inequality sign. When dealing with inequalities, you perform arithmetic operations such as addition, subtraction, multiplication, or division to isolate the variable.

However, a crucial rule to remember is that if you multiply or divide an inequality by a negative number, you must reverse the direction of the inequality sign. This is different from solving equations and can sometimes be the step that causes confusion.

For absolute value inequalities, like \(|3x - 1| > 11\), recognizing that the absolute value signifies distance on a number line is key. This means the expression inside the absolute value can be greater than a positive threshold or less than the negative threshold. Thus, we split the inequality into two potential situations:

• \(3x - 1 > 11\)
• \(3x - 1 < -11\)

Once solved separately, these inequalities provide two distinct ranges of possible solutions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. These expressions are fundamental since they serve as the building blocks for solving equations and inequalities.

In absolute value inequality problems, the algebraic expression inside the absolute value must be carefully analyzed. For \(|3x - 1| > 11\), the expression "***3x - 1***" requires manipulation to find solutions that meet the condition given by the absolute value.

This manipulation includes simple arithmetic adjustments to both sides of the inequality. The process typically involves:

• Adding or subtracting numbers to isolate terms with variables.
• Dividing or multiplying to solve for the variable.

Understanding these steps is vital, as each step transforms the inequality to maintain equality (or inequality) throughout. Being adept with algebraic expressions and their manipulation empowers students to tackle a wide array of mathematical problems.