Problem 20 - Calculus: One and Several Variables

Question

Solve the inequality. Express the solution as an interval or as the union of intervals. Mark the solution on a number line. $$|3 x-2| \geq 4$$

Step-by-Step Solution

Answer

The solution to the inequality \(|3x - 2| \geq 4\) is \((-∞, -2/3] ∪ [2, ∞)\).

Step 1: Define the positive and negative cases

We must split the inequality into two cases: one where \(3x - 2\) is positive or equal to zero, and the other where it's negative. This gives us two inequalities \((3x - 2) \geq 4\) and \(-(3x - 2) \geq 4\)

Step 2: Solve the first inequality

Solve the first inequality, \((3x - 2) \geq 4\). Add 2 to both sides to get \(3x \geq 6\). Then divide both sides by 3 to isolate x and get \(x \geq 2\) as the solution for the first inequality.

Step 3: Solve the second inequality

Solve the second inequality, \(-(3x - 2) \geq 4\). Distribute the negative sign to get \(-3x + 2 \geq 4\). Subtract 2 from both sides to get \(-3x \geq 2\). Finally, divide by -3 (remembering to reverse the inequality sign since we divided by a negative number), giving us \(x \leq -2/3\) as the solution for the second inequality.

Step 4: Combine the solutions and express the solution as an interval

Our solutions from step 2 and step 3 were \(x \geq 2\) and \(x \leq -2/3\). Because 'x' cannot simultaneously be greater than 2 and less than -2/3, the combined solution is the union of these two intervals: \((-∞, -2/3] ∪ [2, ∞)\).

Step 5: Mark the solution on a number line

On a number line, we mark the points -2/3 and 2. These two points divide the number line into three regions. The regions \(x > 2\) and \(x < -2/3\) are shaded to indicate that they are part of the solution.

Key Concepts

absolute value inequalities

Absolute value inequalities involve an expression within absolute value bars, like \(|3x - 2|\), being compared to a number. To solve such inequalities, we need to consider both positive and negative scenarios for what's inside the absolute value bars.
Here's why: the absolute value of a number represents its distance from zero on the number line, disregarding its sign. So, both positive and negative values can satisfy the inequality.
When you have \(|3x - 2| \geq 4\), you essentially split the inequality into two parts:

• The expression itself equal or greater than 4: \(3x - 2 \geq 4\)
• The expression multiplied by -1 being equal or greater than 4: \(-(3x - 2) \geq 4\)

These setups are the two "cases" you solve separately. Each case reflects the different possible scenarios for the value inside the absolute value bars.

interval notation

Interval notation gives us a concise way to express solution sets of inequalities. For the inequality \(|3x - 2| \geq 4\), our solutions describe two ranges on the number line that are not contiguous but cover all numbers that satisfy the inequality.
Instead of writing complex expressions, we use parentheses and brackets:

• Brackets \([ ]\) indicate that an endpoint is included in the interval, meaning 'greater than or equal to' or 'less than or equal to'.
• Parentheses \(( )\) signify that the endpoint is not included, often used for infinity because infinity itself isn't a finite number.

For this problem, the solution set is expressed as \((-\infty, \frac{-2}{3}] \cup [2, \infty)\), indicating all numbers less than or equal to \(-\frac{2}{3}\) and all numbers greater than or equal to 2, with no numbers in between included in the solution.

number line representation

Representing inequalities on a number line provides a visual understanding of the solution set, making it easier to grasp which sections of the number line satisfy the inequality.
Once you have the intervals from the solution, such as \((-\infty, \frac{-2}{3}] \cup [2, \infty)\), you place markers at -\(\frac{2}{3}\) and 2.
These points divide the number line into segments.
The segments that are part of the solution set are then highlighted, often with shading.

• Shading to the left from -\(\frac{2}{3}\) to negative infinity indicates that all values less than or equal to -\(\frac{2}{3}\) satisfy the inequality.
• Shading to the right from 2 to positive infinity indicates that all values greater than or equal to 2 also satisfy the inequality.

This visual cue helps clarify the solution, showing which parts of the number line are included and excluded in the solution.