Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

\(\left\\{\begin{array}{l}|x| \geq 2 \\ |y|<3\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is \((x, y)\) where \(x \geq 2\) or \(x \leq -2\), and \(-3 < y < 3\).

Step 1: Understanding the Absolute Values

The inequality \(|x| \geq 2\) means that the value of \(x\) must either be greater than or equal to 2 or less than or equal to -2. For the inequality \(|y| < 3\), the value of \(y\) must be greater than -3 but less than 3.

Step 2: Solving for x

From the inequality \(|x| \geq 2\), we derive two separate inequalities: \(x \geq 2\) or \(x \leq -2\). This means that \(x\) is either in the interval \([2, \infty)\) or \((-\infty, -2]\).

Step 3: Solving for y

From the inequality \(|y| < 3\), we have that \(-3 < y < 3\). This describes an interval for \(y\) with boundaries at \(-3\) and \(3\), excluding the values \(-3\) and \(3\) themselves.

Step 4: Determine the Combined Solution

The solution to the system is the combination of the solutions of the two inequalities. Thus, for \( x \), we have two intervals \([2, \infty)\) or \((-\infty, -2]\), and for \( y \), we have the interval \((-3, 3)\). The overall solution is the set of points \((x, y)\) where \(x\) is in either \([2, \infty)\) or \((-\infty, -2]\) and \(y\) is in \((-3, 3)\).

Key Concepts

Interval Notation

Interval notation is a concise way of representing a set of numbers, and it is especially useful in mathematics to describe solutions to inequalities. There are two types of intervals: closed intervals and open intervals.
Closed intervals, such as \([a, b]\), include the endpoints \(a\) and \(b\). This means \(x\) can be equal to both \(a\) and \(b\). On the other hand, open intervals do not include the endpoints, like \(a, b\), meaning \(x\) must be greater than \(a\) and less than \(b\).

Additionally, there are half-open intervals, or mixed intervals, where one endpoint is included, such as \([a, b)\) or \(a, b]\).

For the inequalities in the exercise, several intervals are used:

• \([2, \infty)\) refers to numbers greater than or equal to 2 up to infinity.
• \((-\infty, -2]\) encompasses numbers less than or equal to -2 down to negative infinity.
• The interval \((-3, 3)\) involves values greater than -3 and less than 3, excluding both -3 and 3.

Understanding these intervals can help us precisely express ranges of solutions in a clear and mathematical way.

Systems of Inequalities

A system of inequalities consists of two or more inequalities that need to be solved together. When we talk about systems of inequalities, we are looking to find solutions that meet all the given conditions at the same time.
In the context of our exercise, the system involves the inequalities \(|x| \geq 2\) and \(|y| < 3\). This means that any solution must satisfy both conditions.
To solve this:

• First, address each inequality separately as we did, determining the possible values for \(x\) and \(y\) individually.
• Then, combine the results to find pairs of \((x, y)\) that are solutions to the whole system.

Combining the intervals of \(x\) and \(y\) gives us the solution set where the values of \(x\) and \(y\) simultaneously satisfy both inequalities. This is the realm where their conditions overlap.

Solution Sets

Solution sets are the collections of values that fulfill the conditions of a given problem or system. For systems of inequalities, the solution set comes from determining which values satisfy all the inequalities simultaneously.
Returning to the example, our solution involves all \((x, y)\) pairs that fit the criteria:

• \(x\) must belong to the intervals \([2, \infty)\) or \((-\infty, -2]\), thus \(x\) must be outside the interval \((-2, 2)\).
• \(y\) must be within the interval \((-3, 3)\), meaning \(y\) cannot be -3 or 3 itself, but any value in between.

Combining these, the solution set is the collection of all points \( (x, y) \) that meet these criteria. Visualization on a number line or coordinate plane can be particularly helpful, as it graphically represents how \(x\) and \(y\) relate within the given constraints. This visualization aids in better understanding the interaction between these inequalities and their combined solution.