Problem 43 - Algebra and Trigonometry

Question

\(23-48\) 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 is \( (-\infty, -\frac{1}{2}) \cup (\frac{1}{3}, \infty) \).

Step 1: Eliminate Fraction Coefficient on Inequality

Multiply both sides of the inequality \( \frac{1}{2}\left|4x + \frac{1}{3}\right| > \frac{5}{6} \) by 2 to eliminate the fraction: \( \left|4x + \frac{1}{3}\right| > \frac{5}{3} \).

Step 2: Solve the Absolute Value Inequality

The inequality \( \left|A\right| > B \) resolves into two inequalities: \( A > B \) and \( A < -B \). Here, it's \( 4x + \frac{1}{3} > \frac{5}{3} \) or \( 4x + \frac{1}{3} < -\frac{5}{3} \).

Step 3: Solve the First Inequality

For \( 4x + \frac{1}{3} > \frac{5}{3} \), subtract \( \frac{1}{3} \) from both sides: \( 4x > \frac{5}{3} - \frac{1}{3} \). Simplifying gives \( 4x > \frac{4}{3} \). Divide by 4: \( x > \frac{1}{3} \).

Step 4: Solve the Second Inequality

For \( 4x + \frac{1}{3} < -\frac{5}{3} \), subtract \( \frac{1}{3} \) from both sides: \( 4x < -\frac{5}{3} - \frac{1}{3} \). Simplifying gives \( 4x < -\frac{6}{3} = -2 \). Divide by 4: \( x < -\frac{1}{2} \).

Step 5: Express Solution in Interval Notation

The solution is the union of the two intervals from the inequalities: \( x < -\frac{1}{2} \) or \( x > \frac{1}{3} \). In interval notation, this is \( (-\infty, -\frac{1}{2}) \cup (\frac{1}{3}, \infty) \).

Key Concepts

Absolute Value

The absolute value is a mathematical concept that measures the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, like this: \( \left| x \right| \). In essence, it represents the non-negative value of a number. When solving absolute value inequalities, such as \( \left| A \right| > B \), you must consider two possible cases because absolute values can represent both positive and negative situations:

• \( A > B \)
• \( A < -B \)

This means that the inequality handily breaks down into two separate inequalities, allowing all possibilities to be considered. For example, in the context of the provided exercise, \( \left| 4x + \frac{1}{3} \right| > \frac{5}{3} \) becomes two inequalities: \( 4x + \frac{1}{3} > \frac{5}{3} \) and \( 4x + \frac{1}{3} < -\frac{5}{3} \). By solving each of these, we can better understand the range of values for \( x \) that will satisfy the original inequality.

Interval Notation

Interval notation is a concise way of expressing a set of numbers along a number line. It is especially useful when representing the solutions of inequalities.Intervals are written as a pair of endpoints, separated by a comma, within either parentheses "( )" or brackets "[ ]":

• Parentheses \((a, b)\) indicate that a and b are not included in the interval (open interval).
• Brackets \([a, b]\) indicate that a and b are included in the interval (closed interval).

For example, \((-\infty, -\frac{1}{2})\cup(\frac{1}{3}, \infty)\) signifies all numbers less than -1/2 or greater than 1/3. In this notation, \(-\infty\) and \(\infty\) are always paired with parentheses because infinity cannot be included in an interval. In the given exercise, after solving the inequalities derived from the absolute value, the solution was expressed using interval notation as \((-\infty, -\frac{1}{2})\cup(\frac{1}{3}, \infty)\). This notation beautifully showcases all possible solutions to the inequality in a streamlined format.

Fraction Coefficients

Fraction coefficients may initially appear daunting, yet they play a significant role in various mathematical calculations, including solving inequalities. A fraction coefficient is simply a fraction that is applied as a multiplier to a variable in an equation or inequality. To eliminate fraction coefficients, a practical first step is to multiply the entire inequality by the reciprocal of that fraction. This action simplifies the inequality, transforming it into a more workable form. In the problem \( \frac{1}{2}\left|4x + \frac{1}{3}\right| > \frac{5}{6} \), the inequality is made simpler by multiplying both sides by 2 to clear the fraction coefficient \( \frac{1}{2} \). This step changed the inequality to \( \left|4x + \frac{1}{3}\right| > \frac{5}{3} \), making it easier to solve. By removing complex fraction coefficients early, the problem becomes more manageable, allowing us to focus on the core algebraic steps needed to find the solution.