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Chapter 1: Problem 52

Graph the solution set and give the interval notation equivalent. \(x \geq-8\)

Short Answer

Expert verified
The solution set is \([-8, \infty)\).

Step by step solution

01

Understand the Inequality

The given inequality is \(x \geq -8\). This means that we are looking for all values of \(x\) that are greater than or equal to \(-8\).
02

Identify Solution on Number Line

For \(x \geq -8\), start at \(-8\) on the number line. Since \(x\) includes \(-8\), use a closed circle on \(-8\) to represent that \(-8\) is part of the solution set.
03

Shade the Region

Shade the entire region to the right of \(-8\) on the number line. This represents all the numbers greater than \(-8\).
04

Write the Interval Notation

The interval notation for the solution \(x \geq -8\) is \([-8, \infty)\). This indicates that the solutions start at \(-8\) and extend indefinitely to the right.

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Key Concepts

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

Solution Set
The concept of a solution set is essential in solving inequalities. When we have an inequality like \(x \geq -8\), the solution set includes all possible values of \(x\) that make the inequality true. In this case, the solution set starts at \(-8\) because the inequality symbol \(\geq\) signifies that \(x\) can be equal to or greater than \(-8\).

Visualizing a solution set helps us understand which numbers satisfy the inequality. For learners just diving into this concept, it can be helpful to think of the solution set as a list of numbers, even though there are infinitely many numbers within the range. Every number greater than or equal to \(-8\) belongs to our solution set.
Number Line
A number line is a valuable tool for graphing inequalities. It represents numbers spatially along a straight line, allowing us to easily visualize the range of solutions for an inequality.

For the inequality \(x \geq -8\), you start by locating the number \(-8\) on the number line. The entire region to the right of \(-8\) is shaded to indicate all the numbers greater than \(-8\). This shading visually represents our solution set, helping us intuitively understand the range of possible solutions.
  • This simple drawing helps solidify the concept of inequalities for students, making abstract ideas more concrete.
Interval Notation
Interval notation is a mathematical shorthand used to describe a set of numbers along a number line. It's concise and unambiguous.

The interval notation for \(x \geq -8\) is \([-8, \infty)\). The square bracket \([\) next to \(-8\) indicates that \(-8\) is included in the solution set, while the round parenthesis \()\) next to \(\infty\) indicates that it is not a specific number (just the idea of continuous extension).
  • Using interval notation is often faster and clearer than describing a set of numbers with words.
  • It鈥檚 widely used in mathematics to signal an understanding of ranges and limits.
Closed Circle
When graphing inequalities on a number line, the type of circle we use can tell us a lot. A closed circle indicates inclusion of the endpoint.

In the inequality \(x \geq -8\), the closed circle drawn at \(-8\) shows that \(-8\) is part of the solution set. This means we consider \(-8\) as a valid solution for \(x\).
  • Open circles, by contrast, would indicate that the number is not included (for example, in \(x > -8\)).
  • The visual representation with circles helps clarify the difference between \(>\) and \(\geq\) or between \(<\) and \(\leq\).
By using a closed circle, students can see that the inequality includes the boundary point, leading to a deeper understanding of how to read and graph inequalities.

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