Problem 30 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Give the solution set using interval notation. $$|3 x-4| \geq 2$$

Step-by-Step Solution

Answer

The solution set is \((-\infty, \frac{2}{3}] \cup [2, \infty)\).

Step 1: Understand the Absolute Value Inequality

We need to solve the inequality \(|3x - 4| \geq 2\). Absolute value inequalities can be split into two separate inequalities: \(3x - 4 \geq 2\) or \(3x - 4 \leq -2\).

Step 2: Solve the First Inequality

Start with the first part of the inequality: \(3x - 4 \geq 2\). Add 4 to both sides to obtain \(3x \geq 6\). Then, divide both sides by 3 to get \(x \geq 2\).

Step 3: Solve the Second Inequality

Now solve the second part of the inequality: \(3x - 4 \leq -2\). Add 4 to both sides to get \(3x \leq 2\). Then, divide both sides by 3 to get \(x \leq \frac{2}{3}\).

Step 4: Combine the Solution Sets

Combine the solutions of the two inequalities. Since the inequality is 'greater than or equal to,' the solution set includes all values where \(x \geq 2\) or \(x \leq \frac{2}{3}\). In interval notation, this is written as \((-\infty, \frac{2}{3}] \cup [2, \infty)\).

Key Concepts

inequality solving

Solving inequalities involves finding the range of values that satisfies the given condition. In this exercise, we are working with an absolute value inequality \( |3x - 4| \geq 2 \). Absolute value inequalities like this one can be split into two different but connected inequalities.

Here, we consider two cases:

• The expression inside the absolute value is positive or zero:
  \( 3x - 4 \geq 2 \).
  
  By solving, we get:
  \( 3x \geq 6 \implies x \geq 2 \)


• The expression inside the absolute value is negative:
  \( 3x - 4 \leq -2 \).
  
  By solving, we get:
  \( 3x \leq 2, \implies x \leq \frac{2}{3} \)

This means we need to consider both inequalities independently. Finally, combining these results provides the final solution set in terms of inequalities.

interval notation

Interval notation is a way to represent the solution set of an inequality in a more concise form. For our inequality \( |3x - 4| \geq 2 \), we have two sets of values:

• Values of x where
  \( x \geq 2 \)
• Values of x where
  \( x \leq \frac{2}{3} \)

In interval notation, these are written as:

• \( [2, \infty) \): representing the numbers starting from 2 and going to infinity


• \( (-\infty, \frac{2}{3}] \): representing the numbers from negative infinity up to and including \frac{2}{3}

The combined solution, using the union symbol \( \cup \), is: \( (-\infty, \frac{2}{3}] \cup [2, \infty). \)

absolute value operations

Understanding absolute value is key when solving inequalities. Absolute value represents the distance of any number from zero on the number line, regardless of direction. So, for any value x:

\( |x| = \begin{cases} x, & \text{{if }} x \geq 0 \ \ -x, & \text{{if }} x < 0 \end{cases} \)

When we say \( |3x - 4| \geq 2 \), it means:

• Either, \( 3x - 4 \geq 2 \),
  which leads to solving \( 3x - 4 \geq 2, \text{{giving us }} x \geq 2 \)


• Or, \( -(3x - 4) \geq 2 \) or equivalently
  \( 3x - 4 \leq -2 \), which leads to solving \( 3x - 4 \leq -2, \text{{giving us }} x \leq \frac{2}{3} \)

In both cases, we split the absolute value inequality into two separate inequalities and solve them individually. This method is essential for understanding and solving any type of absolute value inequality.