Problem 7 - Elementary Algebra Within Reach

Question

Write a verbal description of the inequality and sketch its graph. $$5 \leq z \leq 10$$

Step-by-Step Solution

Answer

The verbal description for the inequality \(5 \leq z \leq 10\) is: 'z is a number that is at least 5, but no more than 10'. The graph would be a number line with a closed circle at 5 and 10, and a line segment connecting these two points.

Step 1: Understand the Inequality

The inequality \(5 \leq z \leq 10\) is a compound inequality with two parts: \(5 \leq z\) and \(z \leq 10\). This means that z is greater than or equal to 5 AND at the same time, z is less than or equal to 10. In other words, z is a number between 5 and 10, inclusive.

Step 2: Define the verbal description

The verbal description could be: 'z is a number that is at least 5, but no more than 10'.

Step 3: Graph the Inequality

Start by drawing a number line, then mark the points 5 and 10 on the number line. Since z is equal to both 5 and 10, make the marks at these points as closed circles. Then, draw a line segment between these two points, because 'z' includes all the numbers between 5 and 10.

Key Concepts

Graphing Inequalities

Graphing inequalities on a number line is a useful way to visualize the solutions of an inequality. When dealing with compound inequalities, such as the example given \(5 \leq z \leq 10\), you need to understand that you are dealing with two connected inequalities. This means the values of \(z\) satisfy both inequalities simultaneously.

To graph this on a number line:

• Draw a horizontal line (this is your number line).
• Identify and mark the critical points (in this case, 5 and 10).
• If the inequality includes the endpoints (\(5\) and \(10\) in this example), represent these with closed circles, indicating that these values are part of the solution set.
• Draw a solid line (or thick line) between the closed circles at 5 and 10 to show that any number between these two is a solution.

This graphical representation helps you to easily see all potential values of \(z\) that satisfy the inequality.

Verbal Descriptions

Verbal descriptions translate mathematical inequalities into plain language, making them easier to understand. For the compound inequality \(5 \leq z \leq 10\), a verbal description can simplify your understanding of what the inequality conveys.

In this case, the verbal description would be:

• '\(z\) is a number at least 5, but no more than 10.'

This clearly outlines that \(z\) includes the numbers 5 and 10 as well as any number in between them.

Using verbal descriptions in math is helpful in communicating the range or conditions that apply to a variable without having to interpret mathematical symbols immediately. It's a bridge between the abstract math world and everyday language.

Number Line Representation

A number line is a simple yet powerful tool for representing inequalities. It visually displays which numbers satisfy the conditions set by an inequality.

When representing \(5 \leq z \leq 10\) on a number line:

• Draw the number line with key numbers marked. For this inequality, you must identify 5 and 10.
• Mark the endpoints, 5 and 10, with closed circles. These closed circles indicate that the endpoints are included in the solution set.
• Draw a line connecting the two closed circles. This indicates that all numbers between 5 and 10 are also possible values for \(z\).

This visual format makes it easy to see which values are included in the solution, providing a clear and intuitive way to understand inequalities.