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Chapter 8: Problem 48

Graph and write interval notation for each compound inequality. $$ 0 \leq y \leq 4 $$

Short Answer

Expert verified
The interval notation is \([0, 4]\) and the graph includes the numbers from 0 to 4 with filled circles at 0 and 4 and a shaded area in between.

Step by step solution

01

- Understand the Inequality

The inequality given is compound and reads: \(0 \leq y \leq 4\). This represents all values of \(y\) that are greater than or equal to 0 and less than or equal to 4.
02

- Graph the Inequality

To graph this inequality, draw a number line and mark the points at 0 and 4. Since the inequality includes the endpoints (0 and 4), use filled circles to indicate that these values are included. Shade the region between and including 0 and 4.
03

- Write the Interval Notation

The interval notation for the compound inequality \(0 \leq y \leq 4\) is written as \([0, 4]\). The square brackets indicate that the endpoints are included in the interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Inequalities
Graphing inequalities is a vital concept in mathematics. It allows us to visually represent the range of solutions an inequality can have. When graphing an inequality like \(0 \leq y \leq 4\), you need to follow these steps:
  • Draw a number line: This is the base of your graph where you'll mark key points.
  • Identify critical points: For the given inequality, these are 0 and 4.
  • Determine inclusion or exclusion: Since the inequality includes the endpoints, marked as \(0 \leq y \leq 4\), you'll use filled circles at 0 and 4. A filled circle indicates that the number is included in the range.
  • Shade the region: You'll shade the part of the number line between 0 and 4, also including the endpoints. This shading represents all the values that \y\ can take between and including 0 and 4.

Graphing properly helps visualize where the solution set lies.
Interval Notation
Interval notation is another way to express the range of values in an inequality. It uses brackets and parentheses to show which numbers are included or excluded.
For the inequality \(0 \leq y \leq 4\):
  • Identify the endpoints: The start (0) and the end (4) of the range.
  • Determine inclusion: Since 0 and 4 are included in the inequality \(0 \leq y \leq 4\), we use square brackets [ ] to show inclusion.

Therefore, the interval notation is written as \([0, 4]\). This notation compactly shows that \(y\) can take any value from 0 to 4, inclusive. Using interval notation is efficient and precise, providing a clear way to write inequalities.
Number Line
A number line is a fundamental tool in mathematics to visualize numbers and their relationships.
Here's how to use a number line for inequalities like \(0 \leq y \leq 4\).
  • Draw a straight horizontal line: This represents all real numbers.
  • Mark the critical points: Identify and place points 0 and 4.
  • Determine inclusion with filled circles: Since the inequality includes the boundary points, place filled circles at 0 and 4.
  • Shade the relevant region: Shade the area between and including 0 and 4. This shows that \(y\) can be any value within this segment.

By clearly using a number line, it's easy to see the range and limits of the inequality.

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Most popular questions from this chapter

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