Problem 11 - Algebra A Combined Function

Question

Graph each inequality on the number line. $$ 0 \leq y<2 $$

Step-by-Step Solution

Answer

Graph includes 0 (solid dot) to just before 2 (open circle).

Step 1: Understand the Inequality

The given inequality is \(0 \leq y < 2\). This means \(y\) is greater than or equal to 0, and less than 2. Our task is to graph this range on a number line.

Step 2: Draw the Number Line

Draw a horizontal line to represent the number line. Label the key points that are relevant to the inequality, such as 0, 1, and 2.

Step 3: Identify the Range for \(y\)

For \(0 \leq y\), interpret this as the interval starting from 0, including 0 (since it's "greater than or equal to"). For \(y < 2\), understand this as stopping just before 2 (since it's "less than").

Step 4: Plot the Interval

On the number line, draw a solid dot at 0 to indicate that this value is included in the interval. Then, draw an open circle at 2 to show that this value is not included. Connect these points with a line segment to represent all the values \(y\) can take between 0 and 2.

Key Concepts

Number Line

A number line is a straight, horizontal line used to visually represent numbers in order. It's fundamental for graphing inequalities and understanding numerical relationships. On a number line, every point corresponds to a real number.

To graph inequalities like our example, we first draw a horizontal line. It should include evenly spaced markings that represent the key numbers of interest. In the inequality of our exercise, the critical numbers are 0, 1, and 2. These help us clearly indicate where the solutions lie.

The number line helps students visualize which values satisfy the inequality by highlighting the region where the inequality's condition is true. It serves as a map, showing all possible values that a variable, such as \(y\), could assume based on the inequality constraints.

Inequality Interval

An inequality interval is a way of representing all numbers between two specific points that satisfy an inequality. Using inequality notation, it communicates which numbers a variable can take.

Let's break down the exercise's inequality: \(0 \leq y < 2\). It comprises two parts:

• \(0 \leq y\): This signifies that \(y\) includes all values starting from zero and going higher. The 'less than or equal to' part means zero is part of this set.


• \(y < 2\): Here, \(y\) takes values that are less than 2. The strict 'less than' implies 2 itself is not included in this set.

Combining these, the inequality interval for \(y\) includes all numbers from 0 up to, but not including, 2.

Inequality intervals are particularly useful in grasping which range of numbers a variable can span, aiding in both theoretical comprehension and practical graphing tasks.

Closed and Open Circles

When graphing inequalities on a number line, closed and open circles succinctly indicate whether a boundary value is included or excluded from the solution set.

Closed Circle: Used when the inequality involves "less than or equal to" (\(\leq\)) or "greater than or equal to" (\(\geq\)). In our example, \(0 \leq y\), we use a closed circle at 0. This shows that the value "0" is part of the solution, as \(y\) can be equal to 0.

Open Circle: Applied when the inequality uses "less than" (<) or "greater than" (>). For \(y < 2\), an open circle at 2 denotes that the number "2" itself is not included in the interval.

These visual markers make it crystal clear at what exact points the inequality starts and stops being valid. By employing closed and open circles, you effectively communicate inclusivity and exclusivity at these specific boundary values on a number line.