Problem 29 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Give the solution set using interval notation. $$|2 x+5| \geq 3$$

Step-by-Step Solution

Answer

The solution set in interval notation is \ \left( -\infty, -4 \right] \cup \left[ -1, \infty \right) \.

Step 1: Understand the absolute value inequality

The given inequality is \( |2x + 5| \geq 3 \). Absolute value inequalities of the form \( |A| \geq B \) can be split into two separate inequalities: \( A \geq B \) or \( A \leq -B \).

Step 2: Split the inequality

Split the inequality into two parts: \( 2x + 5 \geq 3 \) or \( 2x + 5 \leq -3 \).

Step 3: Solve the first inequality

Solve \( 2x + 5 \geq 3 \): \ 2x \geq -2 \, then divide by 2 to get \ x \geq -1 \.

Step 4: Solve the second inequality

Solve \( 2x + 5 \leq -3 \): \ 2x \leq -8 \, then divide by 2 to get \ x \leq -4 \.

Step 5: Combine the solution sets

The solution set is the combination of both inequalities: \ x \leq -4 \) or \( x \geq -1 \). In interval notation, this is written as \( \left( -\infty, -4 \right] \cup \left[ -1, \infty \right) \.

Key Concepts

interval notation

Interval notation is a concise way of representing sets of numbers, often used to describe solution sets for inequalities.
Instead of listing all the values within a range, interval notation uses parentheses and brackets to denote the start and end points of an interval.
Here are some key points:

• Parentheses ( ) indicate that the endpoint is not included (open interval).
• Brackets [ ] indicate that the endpoint is included (closed interval).
• A combination like (-∞, 5] means all values less than or equal to 5.
• Union (∪) is used to combine multiple intervals, indicating that the solution set includes values from either interval.

Remembering these rules can help you accurately express the solutions to inequalities using interval notation.

splitting inequalities

When dealing with absolute value inequalities like |2x + 5| ≥ 3, the process of 'splitting' the inequality is crucial.
Absolute value represents the distance from zero, so when it is greater than or equal to a number, the expression inside can be either positive or negative.
Here's how you can split it:

• First, understand the form: |A| ≥ B translates to A ≥ B or A ≤ -B.
• In our example, |2x + 5| ≥ 3 gets split as 2x + 5 ≥ 3 or 2x + 5 ≤ -3.
• Solving these two separate inequalities provides the ranges where the original inequality holds true.

Splitting the inequalities in this way enables us to identify distinct solution sets, leading to a comprehensive and correct answer.

solution sets

Solution sets represent the values that satisfy a given inequality or equation.
At the end of solving an absolute value inequality, you combine the results from the 'splitting' process to form the final solution set.
Here's how to interpret solution sets:

• After splitting |2x + 5| ≥ 3, we found x ≥ -1 or x ≤ -4.
• The solution set is all values that make at least one of these inequalities true.
• In interval notation, this is written as (-∞, -4] ∪ [-1, ∞), meaning values less than or equal to -4 or greater than or equal to -1.

Understanding solution sets and expressing them correctly in interval notation ensures clarity and accuracy in mathematical communication.