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Chapter 2: Problem 5

Match each inequality in Column I with the correct graph or interval in Column II. $$ -3

Short Answer

Expert verified
E. (-3, 3)

Step by step solution

01

- Understand the Inequality

The given inequality is -3 < x < 3.It states that x can be any value between -3 and 3, exclusive of -3 and 3.
02

- Analyze the Options

Look at the intervals in Column II:C. (3, 鈭�)D. (-鈭�, 3]E. (-3, 3)F. [-3, 3]We need to match the correct interval with the given inequality.
03

- Identify the Correct Interval

The only interval that matches -3 < x < 3 is option E: (-3, 3). This interval means x is greater than -3 and less than 3, which matches the inequality precisely.

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Key Concepts

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

inequality representation
Linear inequalities represent a range of values that fulfill certain conditions. In the inequality -3 < x < 3, the variable x can take any value between -3 and 3, but not including -3 and 3 themselves. This type of inequality shows:
  • The lower limit (-3) and the upper limit (3) with x in between.
  • Both limits are exclusive, meaning x does not equal -3 or 3.
Inequalities can have different forms:
  • < means less than
  • > means greater than
  • 鈮� means less than or equal to
  • 鈮� means greater than or equal to
interval notation
Interval notation is a concise way to represent the set of solutions for inequalities. For the inequality -3 < x < 3, the interval notation is (-3, 3). Here鈥檚 a breakdown:
  • The parentheses ( and ) indicate that -3 and 3 are not included in the interval.
  • This is also known as an open interval.
In contrast, the interval notation for inequalities that include the endpoints might use square brackets. For example:
  • [-3, 3] includes -3 and 3, known as a closed interval.
  • (-3, 3] includes 3 but not -3; this is a half-open (or half-closed) interval.
  • [3, 鈭�) includes 3 and extends to infinity, where 鈭� indicates boundlessness.
graphing inequalities
Graphing inequalities helps to visually understand the range of solutions. For the inequality -3 < x < 3:
  • Draw a number line.
  • Mark the points -3 and 3 on the line.
  • Use open circles on -3 and 3, indicating that these points are not included.
  • Shade the region between -3 and 3 to show all the possible values x can take.
When you see a graph with shading:
  • Open circles mean the endpoint is not included (exclusive).
  • Closed circles mean the endpoint is included (inclusive).
Using different notations and graphs can make it easier to understand and solve inequalities in various scenarios.

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Most popular questions from this chapter

Answer the questions in Exercises \(5-8 .\) Suppose that you know that two angles of a triangle have equal measures and the third angle measures \(36^{\circ} .\) How would you find the measures of the equal angles without actually writing an equation?

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