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Chapter 4: Problem 30

Graph and write interval notation for each compound inequality. $$ 0 \leq y \leq 5 $$

Short Answer

Expert verified
Interval notation: [0, 5]. Graph: Closed circles at 0 and 5 with shading in between.

Step by step solution

01

Identify the Inequality

The given compound inequality is 0 鈮� y 鈮� 5.
02

Convert to Interval Notation

Convert the compound inequality into interval notation. Since both endpoints 0 and 5 are included in the inequality (indicated by the 鈮� sign), use brackets to denote inclusion. The interval notation is [0, 5].
03

Graph the Inequality

Draw a number line and plot points at 0 and 5. Since both endpoints are included (indicated by the brackets), use closed circles to indicate that 0 and 5 are part of the solution. Shade the region between 0 and 5.

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Key Concepts

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

Interval Notation
Interval notation is a shorthand way of writing inequalities. It shows the range of values that satisfy the inequality. Here's how you represent different types of intervals:
  • Closed Interval: When both endpoints are included, like in \[0, 5\]. This means 0 鈮� y 鈮� 5. Brackets [ ] are used.
  • Open Interval: When neither endpoint is included, like in \(0, 5\). This means 0 < y < 5. Parentheses ( ) are used.
  • Half-Open Interval: When one endpoint is included and the other is not, like in \[0, 5\) or \(0, 5\]. This means one endpoint uses a bracket, while the other uses a parenthesis.
In our exercise, we used a closed interval, \[0, 5\], where both 0 and 5 are included. If you understand these differences, interval notation can make solving and understanding inequalities much easier.
Number Line Graph
Graphing compound inequalities on a number line helps you visualize the solution set. Here's how to do it step-by-step:
  • Draw a horizontal line and mark the points (or numbers) you're interested in. For our example, we mark 0 and 5.
  • If the inequality includes the endpoints (indicated by 鈮� or 鈮�), use closed circles to mark these points. In our exercise, both 0 and 5 are included, so we use closed circles on 0 and 5.
  • Shade the region between the points to show all numbers in between are also solutions. In this case, shade the line segment between 0 and 5 to indicate all values \[0, 5\].
This visual aid makes it clear which parts of the number line are included in the solution.
Inequality Symbols
Understanding inequality symbols is key to solving and graphing inequalities. Let's go over the main ones:
  • \( < \) and \( > \): The inequality symbols for 'less than' and 'greater than'. These do not include the endpoint values. For example, \(y < 5\) means all values less than 5 but not 5 itself.
  • \( \leq \) and \( \geq \): The inequality symbols for 'less than or equal to' and 'greater than or equal to'. These include the endpoint values. In our exercise, 0 \leq y \leq 5 means y can include 0, 5, and all values in between.
Knowing which symbol to use helps you determine whether to use closed or open circles when graphing, and brackets or parentheses in interval notation.

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Most popular questions from this chapter

Waterfalls. In order for a waterfall to be classified as a classical waterfall, its height must be less than twice its crest width, and its crest width cannot exceed one-and-one-half times its height. The tallest waterfall in the world is about 3200 ft high. Let \(h\) represent a waterfall's height, in feet, and w the crest width, in feet. Write and graph a system of inequalities that represents all possible combinations of heights and crest widths of classical waterfalls.

Complete each statement using the word intersection or the word union. The _____ of two sets is the set of all elements that are in either set or in both sets.

Solve. Write the answer using set notation. $$ |x-2|=6 $$

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Solve and graph. Write the answer using both set-builder notation and interval notation. $$ |x-3|+2>7 $$
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