Problem 10 - Algebra A Combined Function

Question

Graph each inequality on the number line. $$ -2 \leq x \leq 3 $$

Step-by-Step Solution

Answer

Shade the number line between -2 and 3, including solid dots at -2 and 3.

Step 1: Identify the Inequality Components

The given inequality is \(-2 \leq x \leq 3\). This can be split into two inequalities: \(-2 \leq x\) and \(x \leq 3\). The first part demonstrates that \(x\) is greater than or equal to \(-2\), and the second part shows that \(x\) is less than or equal to \(3\).

Step 2: Determine the Type of Boundaries

Both parts of the inequality include the equal sign, which indicates that the boundaries \(-2\) and \(3\) are inclusive. On the number line, this is represented by solid dots at \(-2\) and \(3\).

Step 3: Graph the Inequality

To graph \(-2 \leq x \leq 3\) on a number line: 1. Draw a solid dot at \(-2\) to represent that this point is included.2. Draw a solid dot at \(3\) to represent that this point is also included.3. Shade the region between \(-2\) and \(3\) to show all possible values of \(x\) that satisfy the inequality.

Key Concepts

Number Line

The number line is a straight horizontal line on which every point corresponds to a real number. It is a visual representation that allows us to easily understand the relationships between numbers and inequalities. On a number line:

• Points to the left are smaller numbers.
• Points to the right are larger numbers.
• Zero is usually located in the center.

Place of zero can vary if focusing on a specific range.
Spacing between points is uniform to maintain accuracy. When graphing inequalities, the number line becomes a powerful tool for visualizing the range of solutions.

Inclusive Boundaries

In terms of inequalities, inclusive boundaries indicate that the endpoint values are part of the solution set. This type of boundary is shown in inequalities with a 'less than or equal to' (\(\leq\) and greater than or equal to' (\(\geq\)). For example, in the inequality \(-2 \leq x \leq 3\):

• \(-2\) is an inclusive boundary, meaning \(x\) could be \(-2\).
• \(3\) is also an inclusive boundary, meaning \(x\) could be \(3\).

This contrasts with strict inequalities such as \(-2 < x < 3\), where \(x\) can approach \(-2\) and \(3\) but cannot be exactly \(-2\) or \(3\).
On a number line, inclusive boundaries are represented with solid dots to indicate that the endpoint values are included in the solution set.

Graphing Inequalities

Graphing an inequality on a number line is a straightforward process that visually represents the range of solutions. Here's a simple way to accomplish this:

• Identify the endpoints of the inequality. These are the numbers involved in the inequality expression.
• Determine if the boundaries are inclusive or exclusive. As discussed, inclusive boundaries require solid dots, while exclusive ones require open dots.
• Place solid dots on the number line at the inclusive boundaries, \(-2\) and \(3\).
• Shade the area between these dots to represent all possible values of \(x\) that satisfy the inequality.

This shaded region tells us the range of numbers that satisfy the inequality, making it an effective visual method to understand solution sets.