Problem 22 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, -2] \cup [2, \infty)\).

Step 1: Isolate the Absolute Value

Start with the given inequality: \( \frac{1}{2}|x| \geq 1 \). Multiply both sides by 2 to eliminate the fraction: \(|x| \geq 2\).

Step 2: Break into Two Inequalities

Since \(|x| \geq 2\), this can be split into two separate inequalities: 1. \( x \geq 2 \)2. \( x \leq -2 \) These represent the two parts of the absolute value condition being greater than or equal to 2.

Step 3: Write Solution in Interval Notation

Express the solution from Step 2 in interval notation. The solution is the union of the intervals: \((-\infty, -2]\) and \([2, \infty)\).Therefore, the solution in interval notation is: \((-\infty, -2] \cup [2, \infty)\).

Key Concepts

Interval Notation

Interval notation is a way to describe the set of numbers that satisfy an inequality. It's a shorthand method used to express a range of values on the number line. In interval notation, we use parentheses \(()\) and brackets \([]\) to denote whether endpoints are included or excluded.

• Parentheses \(()\): Used when the endpoint is not included in the interval. For example, \((a, b)\) means greater than \(a\) and less than \(b\), but does not include \(a\) or \(b\).
• Brackets \([]\): Used when the endpoint is included. For example, \([a, b]\) means the same as \((a, b)\) but includes \(a\) and \(b\).

When writing interval notation, we also use \(-\infty\) and \(\infty\) to represent all numbers less than any finite number and greater than any finite number, respectively. Importantly, infinity symbols always use parentheses because infinity isn't a specific number that can be reached or included.
For the inequality \(|x| \geq 2\), the interval notation is written as \((-\infty, -2] \cup [2, \infty)\), meaning all values of \(x\) that are either less than or equal to \(-2\), or greater than or equal to \(2\).

Inequality Solution

Solving an inequality, like an equation, involves finding all possible values of the variable that make the inequality true. The key difference is that inequalities can have a range of solutions rather than a single solution.
To solve inequalities effectively:

• Isolate the variable: Just like solving equations, try to get the variable of interest by itself on one side of the inequality.
• Consider the direction: When multiplying or dividing both sides by a negative number, reverse the inequality symbol.
• Check the solution: Substitute values back into the original inequality to verify they work.

In the given problem, isolating the absolute value gives us \(|x| \geq 2\), and from there, we find two separate inequalities: \(x \geq 2\) and \(x \leq -2\). These represent values predicted by the problem, forming the complete solution \((-\infty, -2] \cup [2, \infty)\). By checking, you confirm that all values within these intervals satisfy the initial inequality.

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value symbols, which measure the distance a number is from zero on the number line without considering direction. The expression \(|x|\) indicates how far \(x\) is from zero, always producing a non-negative result.
When dealing with absolute value inequalities like \(|x| \geq 2\), there are two parts to consider because the absolute value can be seen as a "mirror" reflecting values on both positive and negative sides.

• Greater than (or equal): For \(|x| \geq c\), break it into two conditions: \(x \geq c\) and \(x \leq -c\).
• Less than: For \(|x| < c\), the solutions are between \(-c\) and \(c\), forming \(-c < x < c\).

Understanding how to separate the expression into two inequalities is crucial in solving them. It's like finding two sets of numbers on either side of zero that fulfill the given condition.
In solving \(\frac{1}{2}|x| \geq 1\), once you isolate \(|x| \geq 2\), split it into \(x \geq 2\) and \(x \leq -2\). This highlights how you handle absolute values greater than a certain number, effectively utilizing the concept to capture all possible solutions.