Problem 39 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ \frac{1}{2}\left|4 x+\frac{1}{3}\right|>\frac{5}{6} $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, -\frac{1}{2}) \cup (\frac{1}{3}, \infty)\).

Step 1: Isolate the absolute value

First, we want to isolate the absolute value expression. Start by multiplying both sides of the inequality by 2 to eliminate the fraction on the left side.

Step 2: Calculation for Step 1

\[\left|4x + \frac{1}{3}\right| > \frac{5}{3}\]

Step 2: Solve the positive case

In this step, we'll solve the inequality for when the expression inside the absolute value is greater than the positive side.\[4x + \frac{1}{3} > \frac{5}{3}\]

Step 4: Calculation for Step 2

Subtract \(\frac{1}{3}\) from both sides:\[4x > \frac{5}{3} - \frac{1}{3} \4x > \frac{4}{3}\]

Step 3: Solve for x in the positive case

Divide both sides by 4:\[ x > \frac{1}{3}\]

Step 4: Solve the negative case

Now solve for when the expression inside the absolute value is less than the negative side.\[4x + \frac{1}{3} < -\frac{5}{3}\]

Step 7: Calculation for Step 4

Subtract \(\frac{1}{3}\) from both sides:\[4x < -\frac{5}{3} - \frac{1}{3} \4x < -\frac{6}{3} \4x < -2\]

Step 5: Solve for x in the negative case

Divide both sides by 4:\[x < -\frac{1}{2}\]

Step 6: Combine the solutions

Since we solved both cases, combine the results into one solution using union notation: \[x < -\frac{1}{2} \text{ or } x > \frac{1}{3}\]

Step 7: Express in interval notation

Translate the combined solution into interval notation:\[(-\infty, -\frac{1}{2}) \cup (\frac{1}{3}, \infty)\]

Key Concepts

Absolute Value

The absolute value is a concept that measures the distance of a number from zero on the number line. It is always non-negative, regardless of whether the number itself is positive or negative. When dealing with absolute values, think of it as asking, "How far is this number from zero?" without considering direction. For example:

• The absolute value of 4 is written as \(|4|\) and equals 4.
• The absolute value of -4 is also \(|-4|\) and equals 4.

In inequalities, the absolute value \( |a| > b \) implies that \( a \) could be greater than \(+b\) or less than \(-b\). This forms two separate inequalities because the expression inside the absolute value can be above or below zero to achieve the obtained result, effectively doubling the number possibilities to solve.

Interval Notation

Interval notation is a method of writing subsets of the real number line. This method is concise and clear, providing an easy way to express the solution set of inequalities. In interval notation, intervals are written with parentheses \( ( \) and \( ) \) or brackets \( [ \) and \( ] \) to describe which numbers are included or excluded, respectively. Let’s break it down:

• Parentheses \( ( \) and \( ) \) denote that the end value is not included (open interval).
• Brackets \( [ \) and \( ] \) denote that the end value is included (closed interval).

For example, \((-\infty, -\frac{1}{2}) \cup (\frac{1}{3}, \infty)\) means all numbers less than \(-\frac{1}{2}\) or greater than \( \frac{1}{3}\). The union symbol \( \cup \) indicates that the solution includes both intervals. This notation is powerful in expressing solutions to inequalities concisely.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a value or set of values. They are the building blocks of algebraic equations and inequalities. In the given exercise, the expression \( 4x + \frac{1}{3} \) is an algebraic expression. Understanding these expressions is key to solving inequalities like the one at hand.When dealing with algebraic expressions:

• Identify and isolate key parts of the expression, such as terms with variables.
• Use inverse operations to simplify or solve these expressions.

For solving inequalities, especially those involving algebraic expressions with absolute values, breaking down and separating the inequality into parts as shown in the steps allows for clear and manageable solutions. Each part corresponds to a condition under which the original expression and its transformations can hold. By doing this, you can effectively find the range of values that satisfy the original inequality.