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Chapter 1: Problem 51

Graph the solution set and give the interval notation equivalent. \(\quad x>-3\)

Short Answer

Expert verified
The graph is an open circle at -3 with shading to the right. Interval: \((-3, \infty)\).

Step by step solution

01

Identify Inequality and Critical Point

The given inequality is \( x > -3 \). The critical point here is \(-3\), which is not included in the solution set since it's a strict inequality (\(>\)).
02

Graph the Inequality

On a number line, place an open circle on \(-3\) to indicate that \(-3\) is not included. Shade all numbers to the right of \(-3\) to represent that \( x \) can be any number greater than \(-3\).
03

Write the Interval Notation

Since the solution includes all numbers greater than \(-3\), the interval notation is \((-3, \infty)\). The parenthesis at \(-3\) indicates it is not included, and \(\infty\) always has a parenthesis because it's not a real number.

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Key Concepts

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

Interval Notation
Interval notation provides a way to describe a range of values for a variable, such as those that satisfy an inequality. For the inequality \( x > -3 \), interval notation helps express that the variable \( x \) can take any value greater than \(-3\).This solution is written as \((-3, \infty)\):
  • The "(" next to \(-3\) means that \(-3\) itself is not included in the interval. This is because the inequality is strict (\( > \) rather than \( \geq \)).
  • "\( \infty \)" symbolizes all numbers greater than any finite number. Because infinity is not a specific point that can be reached, it is always denoted with a parenthesis "(".
Interval notation is compact and avoids wordiness, making it an efficient way to represent solution sets. It is important to know how parenthesis indicate exclusion and brackets indicate inclusion of endpoints.
Number Line Representation
Number lines are a visual tool used for illustrating inequalities and their solutions. By using a number line, we can clearly see which parts of the line satisfy the inequality.For the inequality \( x > -3 \):
  • Start by marking the critical point \(-3\) on the number line with an open circle. This indicates that \(-3\) is not part of the solution.
  • Shade or draw an arrow extending to the right of \(-3\). This visually represents that \( x \) includes any value greater than \(-3\).
Using number line representation helps in confirming the part of the line that matches the inequality expression. This method is particularly useful for anyone who benefits from seeing a picture rather than just words or symbols.
Critical Points in Inequalities
In dealing with inequalities, identifying critical points is a crucial step. A critical point is the specific value that defines where the inequality begins or ends its range.For this inequality \( x > -3 \), the critical point is \(-3\):
  • It separates the values not included in the solution set from those that are.
  • Because it's a strict inequality (\(>\)), the critical point \(-3\) is not included in the solution set. Hence, it is represented by an open circle on the number line.
Understanding critical points helps in defining the boundaries and accurately graphing the inequality. Recognizing whether the inequality is strict (\(<\) or \(>\)) or inclusive (\(\leq\) or \(\geq\)) guides how these points are graphically represented and included in interval notation.

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