Problem 33 - Algebra A Combined Function

Question

Describe the solution of the system: \(\left\\{\begin{array}{l}y \leq 3 \\ y \geq 3\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system is \( y = 3 \).

Step 1: Understand the Inequalities

The system of inequalities consists of two parts: (1) \( y \leq 3 \) and (2) \( y \geq 3 \). This means that \( y \) must be less than or equal to 3 and also greater than or equal to 3.

Step 2: Combine the Inequalities

Since \( y \) needs to satisfy both conditions, we combine the inequalities \( y \leq 3 \) and \( y \geq 3 \). The only number that is both less than or equal to 3 and greater than or equal to 3 is 3 itself.

Step 3: Identify the Solution

The solution to the system of inequalities is the set of all values that satisfy both conditions simultaneously. From Step 2, we concluded that the only number satisfying both inequalities is 3.

Key Concepts

Solving Inequalities

When we talk about solving inequalities, we are referring to finding all the possible values that make an inequality true. Inequalities are similar to equations with one big difference—they tell us that one side can be larger, smaller, or equal to the other, but not necessarily equal.
In this particular exercise, we are solving a system of inequalities. Our system includes two inequalities: \( y \leq 3 \) and \( y \geq 3 \).
This means we are looking for all possible values of \( y \) that satisfy both inequalities. Solving a system of inequalities means identifying the values that work for every inequality in the system.

• Look at each inequality individually.
• Find common values that satisfy all inequalities.

The process usually involves checking all parts to find the solution that makes every inequality true.

Inequalities

Inequalities express a relationship where one value is not necessarily equal to another. Instead, they may be smaller, larger, or sometimes equal. Common symbols used in inequalities include:

• \(<\) meaning "less than"
• \(>\) meaning "greater than"
• \(\leq\) meaning "less than or equal to"
• \(\geq\) meaning "greater than or equal to"

In this exercise, we have two specific inequalities: \( y \leq 3 \) and \( y \geq 3 \).
These inequalities suggest that \( y \) is constrained in its value. The combination of these two particular inequalities tells us that \( y \) can only be exactly 3, because 3 is the only number that satisfies "less than or equal to" and "greater than or equal to" at the same time.

Mathematical Systems

A mathematical system is a collection of equations or inequalities that are solved together. Each equation or inequality adds a condition that the solution must satisfy.
In the case of a system of inequalities, we are often looking for the intersection of the solutions to each inequality. This requires understanding what each inequality implies and knowing how to combine them effectively.
For the resolved system here, we started with two inequalities, \( y \leq 3 \) and \( y \geq 3 \). Both inequalities must be true at the same time for a solution to be valid.
Understanding systems involves:

• Identifying and writing down all the conditions.
• Solving each condition separately, if possible.
• Finding the overlapping solution set that works for all parts of the system.

By combining the two inequalities, the solution was simplified to a single value, showing how systems can narrow down possibilities to precise numbers or ranges.