Problem 36 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ 3-|2 x+4| \leq 1 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, -3] \cup [-1, \infty)\).

Step 1: Simplify the Inequality

Start by subtracting 3 from both sides of the inequality so that the term with the absolute value is isolated on one side:\[-|2x + 4| \leq -2\]This can be rearranged as:\[|2x + 4| \geq 2\]

Step 2: Apply the Definition of Absolute Value Inequality

The inequality \(|2x + 4| \geq 2\) implies two cases, according to the rules of absolute value inequalities:1. \(2x + 4 \geq 2\)2. \(2x + 4 \leq -2\)

Step 3: Solve Case 1

Solve the inequality \(2x + 4 \geq 2\):Subtract 4 from both sides:\[2x \geq -2\]Divide by 2:\[x \geq -1\]

Step 4: Solve Case 2

Solve the inequality \(2x + 4 \leq -2\):Subtract 4 from both sides:\[2x \leq -6\]Divide by 2:\[x \leq -3\]

Step 5: Combine the Solutions

Combine the solutions from steps 3 and 4:From \(x \geq -1\) and \(x \leq -3\), the solution is the union of the two intervals because we are looking for values of \(x\) that satisfy either one of the inequalities.In interval notation, the solution is:\[(-\infty, -3] \cup [-1, \infty)\]

Key Concepts

Absolute Value Inequalities

Absolute value inequalities, such as \(|2x + 4| \geq 2\), are special because they capture the distance from zero. The absolute value of a number is its distance from zero on a number line, regardless of direction. It’s important to remember that when dealing with absolute value inequalities, you have two "halves" to assess: one where the expression inside is positive and one where it's negative.
For the given inequality \(|2x + 4| \geq 2\), it means that the distance between \(2x + 4\) and 0 is at least 2. This splits into two separate inequalities:

• Case 1: \(2x + 4 \geq 2\)
• Case 2: \(2x + 4 \leq -2\)

This dual approach ensures you capture all potential solutions, covering both the positive and negative perspectives on the number line.

Interval Notation

Interval notation is a clean and concise way to express a range of values. It helps us communicate the solutions to inequalities without using words. For example, if we say \(x \geq -1\), in interval notation, it translates to \([-1, \infty)\). This notation uses brackets and parentheses:

• Square brackets [ ] imply that the boundary number is included (i.e., it holds true).
• Parentheses ( ) indicate that the number next to it is not included.

In the solution for the inequality \(|2x + 4| \geq 2\), we have two solution sets: \(x \geq -1\) which is captured by \([-1, \infty)\) and \(x \leq -3\) represented by \((-\infty, -3]\). These intervals are combined using the union symbol \(\cup\), showing all \(x\)-values that meet either condition.

Solving Inequalities Step by Step

Solving inequalities step by step ensures that you don’t miss any critical elements of the solution process. Let's use the inequality \(3 - |2x + 4| \leq 1\) as our guide to learn this approach.
Simplify: This first step removes any constant additional factors around the absolute value, isolating its term. We subtracted 3, getting \(-|2x + 4| \leq -2\) and rearranged as \(|2x + 4| \geq 2\).
Consider Cases: With absolute inequalities, you split them into two linear inequalities.

• In our example, one case was \(2x + 4 \geq 2\).
• The other case was \(2x + 4 \leq -2\).

Solve Each Case: For each, solve like a standard linear inequality: subtract, add, multiply, or divide to isolate \(x\). - Case 1 simplified to \(x \geq -1\), while Case 2 resolved to \(x \leq -3\).
Combine Solutions: Lastly, bring the separate solutions together. This results in intervals \((-\infty, -3]\) and \([-1, \infty)\), uniting them as the overall solution sets using \(\cup\). This step-by-step approach helps ensure clarity and thoroughness, especially with more complex inequalities.