Problem 20 - Intermediate Algebra

Question

\(y \geq-2\)

Step-by-Step Solution

Answer

The solution is \(y \geq -2\) or \([-2, \infty)\).

Step 1: Understanding the Inequality

The inequality given is \(y \geq -2\). This means the values for \(y\) can be any number greater than or equal to \(-2\). Essentially, \(y\) includes \(-2\) and all numbers greater than \(-2\).

Step 2: Representing the Solution

The solution to the inequality can be represented on a number line. For \(y \geq -2\), you would place a solid circle or dot on \(-2\) to indicate that \(-2\) is included in the solution. Then, you would shade the number line to the right of \(-2\) because \(y\) can be any number greater than \(-2\).

Step 3: Writing the Solution Interval

The solution can also be expressed in interval notation. For the inequality \(y \geq -2\), the interval notation is \([-2, \, \infty)\). This notation means all numbers from \(-2\) to infinity, including \(-2\), are solutions.

Key Concepts

Number Line Representation

In mathematics, representing solutions on a number line is a visual way to understand inequalities.
The number line helps us see which numbers satisfy the inequality.
For the inequality \(y \geq -2\), we use a number line to show our solution. Here's how we do it:

• First, locate the number \(-2\) on the number line.
• Place a solid circle or dot on \(-2\). This solid circle indicates that \(-2\) is part of the solution set because the inequality is "greater than or equal to".
• Next, shade the line to the right of \(-2\). This shading represents all the numbers greater than \(-2\) that are also included in the solution set.

By following these steps, anyone can clearly see which values of \(y\) satisfy the inequality \(y \geq -2\).
It's a helpful picture that shows both the boundary and the direction on the number line.

Interval Notation

Interval notation is a concise way to describe a range of values that make an inequality true.
It's often used alongside number line representation to clearly state the solution set.
For the inequality \(y \geq -2\), we use interval notation as follows:

• The bracket \([-2, \) signals that \(-2\) is included in the interval. Square brackets always indicate inclusion.
• The symbol \(\infty\) is used to represent that there is no upper boundary; the numbers continue indefinitely.
• A parenthesis \(\infty)\) is used because infinity is a concept, not a number, and can't be "included" in a set.

Therefore, the interval notation \([-2, \infty)\) represents all real numbers starting from \(-2\) upwards without end.
This makes it easy to understand and communicate the complete set of solutions.

Solution Set

The solution set refers to all possible values that satisfy a given inequality.
In our exercise, the inequality \(y \geq -2\), the solution set consists of every number that meets this condition. To determine the solution set:

• Recognize \(-2\) as the smallest number in the set that satisfies the inequality. It belongs to the solution set because of the \(\geq\) (greater than or equal to) sign.
• All numbers greater than \(-2\) also satisfy the inequality. Thus, they too are part of the solution set.

The solution set is often communicated using number line representation or interval notation, making it evident which values are included.
Understanding the concept of a solution set helps students solve inequalities effectively and interpret them in real-world situations.