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Chapter 2: Problem 52

Graph each compound inequality and describe the graph using interval notation. $$ -1

Short Answer

Expert verified
The graph is between \(-1\) (open circle) and \(4\) (closed circle); interval: \((-1, 4]\).

Step by step solution

01

Understand the inequality

The given compound inequality is \(-1 < x \leq 4\). This means that \(x\) is greater than \(-1\) and less than or equal to \(4\). We interpret this as all values between \(-1\) and \(4\), not including \(-1\) but including \(4\).
02

Set Up the Number Line

Draw a horizontal number line. Identify the points \(-1\) and \(4\) on the line. These are the critical values which define the boundaries for \(x\).
03

Plot the Graph

On the number line, draw an open circle at \(-1\) to indicate that \(-1\) is not included in the solution set. Draw a closed circle at \(4\) to indicate that \(4\) is included. Shade the region between \(-1\) and \(4\) to represent all \(x\) values in this range.
04

Write the Interval Notation

The interval notation for the compound inequality \(-1 < x \leq 4\) is written as \((-1, 4]\). Here, \((-1\) indicates that \(-1\) is not included and \(4]\) indicates that \(4\) is included.

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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 way to express the set of solutions to an inequality in a concise manner. It provides a clear visual representation of which values are included or excluded in a range. In the context of the compound inequality \(-1 < x \leq 4\), interval notation simplifies the expression using parentheses and brackets.

Here鈥檚 how it works:
  • Parentheses, like \( ( \) or \( ) \), denote that a number is not included in the interval. For the inequality above, \(-1\) is not included, so it is written with a parenthesis: \((-1\).
  • Brackets, like \( [ \) or \( ] \), indicate that a number is included in the interval. Since \(4\) is included in the set, use a bracket: \([4\).
Putting it all together, the interval notation \((-1, 4]\) represents all numbers \(x\) such that \(-1 < x \leq 4\). This way of writing helps in quickly understanding the scope of the solution set without having to map out the entire inequality.
Number Line
A number line is a straight, horizontal line that is used to represent numbers and intervals. This is a great visual tool in math to identify ranges and solutions of inequalities. Drawing a number line for the given inequality \(-1 < x \leq 4\) involves a few simple steps:

  • Start by marking the critical points on the line. For this inequality, these are \(-1\) and \(4\), as they establish the boundaries of the solution set.
  • Each point represents an important boundary of the inequality.
The number line helps us to easily visualize which numbers are part of the interval. In this case, you would highlight the segment between \(-1\) and \(4\). The visual helps to reinforce the idea that anything within this segment is valid according to the compound inequality given.
Open and Closed Circles
On a number line, circles are used to indicate whether a certain endpoint or boundary is included in an interval or not. For our inequality \(-1 < x \leq 4\), understanding these circles is key to correctly representing the solution:

  • Open circles are drawn at endpoints that are not part of the solution set. In the interval \(-1 < x\), \(-1\) is not included鈥攈ence, we place an open circle at \(-1\).
  • Closed circles represent endpoints that are included. For \(x \leq 4\), \(4\) is part of the solution, so you use a closed circle at \(4\).
Utilizing open and closed circles helps communicate which specific values are included or excluded and ensures accurate depiction of intervals on a number line. This method is universally recognized and aids in avoiding misunderstandings in interpreting inequalities.

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Most popular questions from this chapter

Multiply. $$ 3(5 t+1) 2 $$

A store warehouses 40 more portables than bigscreen TV sets, and 15 more consoles than big-screen sets. The monthly storage cost for a portable is \(\$ 1.50,\) a console is \(\$ 4.00,\) and a big-screen is \(\$ 7.50 .\) If storage for all the televisions costs \(\$ 276\) per month, how many big-screen sets are in stock?

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ 3(3 x+3)-2 \leq 2(2 x-1)+6 x $$

After solving an equation, how do we check the result?

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ -m-12>15 $$
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