Problem 32 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ \left|\frac{x+1}{2}\right| \geq 4 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -9] \cup [7, \infty)\).

Step 1: Understand the Absolute Value Inequality

The inequality is \( \left|\frac{x+1}{2}\right| \geq 4 \). An absolute value inequality of the form \(|A| \ge B\) can be rewritten as two inequalities: \(A \geq B\) and \(A \leq -B\). In this case, \(A = \frac{x+1}{2}\) and \(B=4\).

Step 2: Set Up the Two Inequalities

According to the rule for absolute value inequalities, we set up the conditions: 1. \(\frac{x+1}{2} \geq 4\) 2. \(\frac{x+1}{2} \leq -4\)

Step 3: Solve the First Inequality

Solve \(\frac{x+1}{2} \geq 4\): Multiply both sides by 2 to eliminate the fraction: \(x+1 \geq 8\) Subtract 1 from both sides: \(x \geq 7\).

Step 4: Solve the Second Inequality

Solve \(\frac{x+1}{2} \leq -4\): Multiply both sides by 2: \(x+1 \leq -8\) Subtract 1 from both sides: \(x \leq -9\).

Step 5: Combine the Solutions

The solution to the original inequality \(\left|\frac{x+1}{2}\right| \geq 4\) is the union of the solutions from the two inequalities: \(x \geq 7\) or \(x \leq -9\).

Step 6: Express the Solution in Interval Notation

The solution, formed by the union of the two intervals, is: \((-\infty, -9] \cup [7, \infty)\).

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. In mathematics, we often use this notation to represent the solution sets of inequalities. Depending on the type of inequality, the interval can be open, closed, or half-open. In an interval:

• Round brackets, \((a, b)\), denote an open interval where endpoints \(a\) and \(b\) are not included.
• Square brackets, \([a, b]\), denote a closed interval where endpoints are included.
• Mixing the brackets like \((a, b]\) or \([a, b)\) denotes a half-open interval.

For the solution \((-\infty, -9] \cup [7, \infty)\), it is made up of two parts or intervals. The \((-\infty, -9]\) part means all numbers less than or equal to \(-9\), and \([7, \infty)\) means all numbers equal to or greater than \(7\). The symbol \(\cup\) indicates the union, representing both intervals as part of the solution.

Inequality Solving

Solving inequalities is similar to solving equations, but with an additional focus on maintaining the inequality direction.

• When multiplying or dividing both sides by a negative number, remember to flip the inequality sign.
• Perform arithmetic operations on both sides like addition or subtraction to isolate the variable.

In our exercise, the absolute value inequality \(\left| \frac{x+1}{2} \right| \geq 4\) was transformed into two separate inequalities: \( \frac{x+1}{2} \geq 4\) and \( \frac{x+1}{2} \leq -4\). To solve each, we first eliminate the fraction by multiplying both sides by 2:
- For \( \frac{x+1}{2} \geq 4\), multiplying gives \(x+1 \geq 8\).- Subtracting 1 from both sides yields \(x \geq 7\).- For \( \frac{x+1}{2} \leq -4\), similarly, multiplication leads to \(x+1 \leq -8\) and subtracting 1 results in \(x \leq -9\).These steps help in isolating the variable and solving each component inequality.

Absolute Value Properties

Absolute value describes the distance of a number from zero on the number line, always being non-negative. Understanding properties of absolute value helps in solving related inequalities. There are specific approaches to handle inequalities with absolute values:

• For \(|A| \ge B\), it means \(A \ge B\) or \(A \le -B\).
• For \(|A| \le B\), it translates to \(-B \le A \le B\).

In our problem, the inequality \(\left| \frac{x+1}{2} \right| \geq 4\) implies that the quantity inside the absolute value can either be greater than or equal to 4 or less than or equal to -4. This leads to two different inequalities that need to be solved separately. These properties allow us to break down and resolve complex absolute value expressions into more manageable linear inequalities.