Math Problem Statement

a. Choose the correct interval notation for the inequality notation x > −3

[-3, ∞)

(-3, ∞)

(-∞ , -3)

(-∞ , 3) None of these

b. Choose the correct interval notation for the inequality notation −3 < x ≤ 3 or 7 ≤ x

(-3, 3] ⋃ [7, ∞)

(-3, 3) ⋃ (7, ∞)

[-3, 3) ⋃ [7, ∞)

[-3, 3] ⋃ (7, ∞) None of these

c. Choose the correct interval notation for the inequality notation x ≤ −3 or 7 ≤ x

(-∞ , -3] ⋃ (7, ∞)

(-∞ , -3) ⋃ (7, ∞)

(-∞ , -3] ⋃ [7, ∞)

(-∞ , 3) ⋃ (7, ∞) None of these

Solution

Let's analyze and solve each part:

a. Choose the correct interval notation for the inequality notation x>3x > -3:

The inequality x>3x > -3 means that xx starts just above 3-3 and goes to infinity. In interval notation, we express this as:

(3,)(-3, \infty)

  • 3-3 is not included, so we use a parenthesis.
  • Infinity \infty is always accompanied by a parenthesis.

Answer for (a): (3,)(-3, \infty)


b. Choose the correct interval notation for the inequality notation 3<x3-3 < x \leq 3 or 7x7 \leq x:

This represents two intervals:

  1. 3<x3-3 < x \leq 3: xx is between 3-3 and 33, but 3-3 is not included while 33 is included.

    • Interval: (3,3](-3, 3]
  2. 7x7 \leq x: xx starts at 7 (included) and goes to infinity.

    • Interval: [7,)[7, \infty)

The union (\cup) of these intervals is: (3,3][7,)(-3, 3] \cup [7, \infty)

Answer for (b): (3,3][7,)(-3, 3] \cup [7, \infty)


c. Choose the correct interval notation for the inequality notation x3x \leq -3 or 7x7 \leq x:

This represents two intervals:

  1. x3x \leq -3: xx starts from negative infinity up to 3-3 (including 3-3).

    • Interval: (,3](-\infty, -3]
  2. 7x7 \leq x: xx starts at 7 (included) and goes to infinity.

    • Interval: [7,)[7, \infty)

The union (\cup) of these intervals is: (,3][7,)(-\infty, -3] \cup [7, \infty)

Answer for (c): (,3][7,)(- \infty, -3] \cup [7, \infty)


Summary of correct answers:

  • a: (3,)(-3, \infty)
  • b: (3,3][7,)(-3, 3] \cup [7, \infty)
  • c: (,3][7,)(-\infty, -3] \cup [7, \infty)

Would you like further details or explanations on any of these solutions?


5 Related Questions:

  1. How do you express strict inequalities in interval notation?
  2. What is the difference between open and closed intervals in inequalities?
  3. How would you express x2x \geq 2 in interval notation?
  4. How do you represent a union of intervals in set notation?
  5. What does an intersection of two intervals mean in terms of inequalities?

Tip:

Always use parentheses for strict inequalities (<< or >>) and brackets for inclusive inequalities (\leq or \geq).

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Math Problem Analysis

Mathematical Concepts

Inequalities
Interval Notation
Set Union

Formulas

No specific formula, but involves interpreting inequalities and intervals

Theorems

No specific theorem, but involves set theory and interval notation rules

Suitable Grade Level

Grades 9-12