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Chapter 1: Problem 57

Graph the solution set and give the interval notation equivalent. \(\quad-12 \leq x \leq-4\)

Short Answer

Expert verified
The interval notation is \([-12, -4]\), and the graph has shaded area from -12 to -4 with closed circles on both ends.

Step by step solution

01

Understanding the Inequality

The given inequality is \(-12 \leq x \leq -4\). This indicates that \(x\) is a value that is greater than or equal to \(-12\) and less than or equal to \(-4\). The solution set includes all real numbers between \(-12\) and \(-4\), inclusive of both endpoints because of the 'equal to' sign.
02

Graphing the Inequality

To represent this inequality on a number line, plot a closed circle at \(-12\) and another closed circle at \(-4\). The closed circles indicate that \(-12\) and \(-4\) are included in the solution set. Shade the entire region between these two points to show that all numbers between \(-12\) and \(-4\) are solutions.
03

Writing in Interval Notation

Interval notation is a way to describe a set of numbers along a number line. The inequality \(-12 \leq x \leq -4\) is written in interval notation as \([-12, -4]\). The square brackets indicate that both endpoints \(-12\) and \(-4\) 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
When it comes to understanding and interpreting algebraic concepts, one fundamental skill is graphing inequalities. This graphical method allows you to visually represent the solution set of an inequality within a specific range of numbers. Consider the inequality \(-12 \leq x \leq -4\).

To graph this inequality:
  • Locate the points \(-12\) and \(-4\) on a number line.
  • Place a closed circle above both \(-12\) and \(-4\). Closed circles demonstrate that these boundary values are part of the solution set.
  • Shade the section of the number line between \(-12\) and \(-4\) to indicate that all numbers within this range, including the endpoints, are solutions to the inequality.
Graphing inequalities using a number line provides a clear and straightforward way to visualize the possible values for \(x\) that satisfy the inequality. It makes it easy to see at a glance which numbers are included in the solution set.
Interval Notation
Interval notation serves as a compact and efficient way to express a range of values, especially when dealing with inequalities. Knowing how to translate an inequality into interval notation is crucial for communicating mathematical ideas clearly and succinctly.

For the inequality \(-12 \leq x \leq -4\), interval notation provides a neat summary. It is written as \([-12, -4]\).

Here's how to understand this representation:
  • The use of square brackets, \([\text{ } ]\), signifies that the endpoints \(-12\) and \(-4\) are included. This is because the inequality includes the 'equal to' component, which allows these boundary numbers within the solution.
Interval notation is particularly helpful in higher-level mathematics, as it offers a quick reference for discussing and writing about continuous ranges of numbers, streamlining communication between students and teachers.
Solution Set in Algebra
The concept of a solution set in algebra involves identifying all possible values that fulfill a given condition or equation. Particularly with inequalities, finding the solution set means determining the range of numbers that satisfy the mathematical statement.

For an inequality like \(-12 \leq x \leq -4\), the solution set includes every number between \(-12\) and \(-4\), inclusive of both these boundaries.
  • The notation for this solution set can be depicted on a number line as shaded areas between the closed circles at \(-12\) and \(-4\).
  • In interval notation, it is expressed as \([-12, -4]\). Each of these methods gives a clear picture of all the numbers that make the inequality true.
By understanding the solution set, students gain insight into the range of values that can satisfy a given expression, enhancing their comprehension of algebraic inequalities and their applications.

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Most popular questions from this chapter

Convert the following temperatures to degrees Celsius given, where \(F\) represents degrees Fahrenheit. \(-40^{\circ} \mathrm{F}\)

Solve. $$ 5(3 x+3)-(10 x-4)=4 $$

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Rewrite using scientific notation. 0.000000231
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