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Questions and Answers
Which of the following is NOT a subset of real numbers?
Which of the following is NOT a subset of real numbers?
Rational numbers can be expressed in the form of a function m/n, where m and n are integers and n ≠ 0.
Rational numbers can be expressed in the form of a function m/n, where m and n are integers and n ≠ 0.
True
The closed interval from a to b is represented as ______.
The closed interval from a to b is represented as ______.
]a, b[
What is the set of x such that x is an integer and a < x < b?
What is the set of x such that x is an integer and a < x < b?
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What is meant by 'solving the inequality'?
What is meant by 'solving the inequality'?
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What is the symbol for an open interval from a to b?
What is the symbol for an open interval from a to b?
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What is the solution set of the inequality 6/(x – 1) ≥ 5 if x > 1?
What is the solution set of the inequality 6/(x – 1) ≥ 5 if x > 1?
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Study Notes
Preliminaries
- Calculus involves real numbers that can be expressed as decimals.
- Real numbers include natural numbers, integers, and rational numbers.
- Natural numbers: 1, 2, 3, 4, ...
- Integers: 0, ±1, ±2, ±3, ...
- Rational numbers: can be expressed as m/n, where m and n are integers and n ≠ 0.
- Examples of rational numbers: 200/67, 13/3, (5 - 5)/67, 1/3, 0/0, 3/0
- Division by zero is undefined.
- Division by zero results in undefined expressions.
- Real numbers can be represented on a number line; this is called the real line.
Intervals
- Interval: a set of real numbers.
- Open interval: (a, b) represents all numbers between a and b, excluding a and b.
- Closed interval: [a, b] represents all numbers between a and b, including a and b.
- Other variations include half-open intervals: (a, b] and [a, b).
Inequalities
- Solving inequalities involves finding the range of numbers that satisfy a specific inequality in the variable 'x.'
- "=" means "implies."
- Inequalities can be solved by manipulating both sides using the following rules:
- Adding or subtracting the same number to both sides doesn't change the solution.
- Multiplying or dividing both sides by the same positive number doesn't change the solution.
- Multiplying or dividing both sides by the same negative number reverses inequality signs.
- Example:
- 2x + 4 < 12 is solved by subtracting 4 from both sides: 2x < 8.
- Then divide both sides by 2: x < 4. Therefore, the solution is all x < 4.
Solving Inequalities
- Example 1:
- 3x + 12 < 0 can be solved by subtracting 12 from both sides: 3x < -12.
- Dividing both sides by 3 gives x < -4.
- The solution set is the open interval (-∞, -4), which means all numbers less than -4.
- Example 2:
- 7x + 3 > -18 can be solved by subtracting 3 from both sides: 7x > -21.
- Then dividing both sides by 7 gives x > -3.
- The solution set is the open interval (-3, ∞), which means all numbers greater than -3.
- Example 3:
- 6/(x - 1) ≥ 5 can hold only if x > 1.
- Multiplying both sides by (x - 1) results in 6 ≥ 5(x - 1).
- This simplifies to 6 ≥ 5x - 5.
- Adding 5 to both sides provides 11 ≥ 5x.
- Dividing both sides by 5 gives x ≤ 11/5.
- The solution set is the half-open interval (1, 11/5], which includes all numbers greater than 1 but less than or equal to 11/5.
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Description
Test your understanding of the basic concepts in calculus, including real numbers, intervals, and inequalities. This quiz covers definitions and examples of natural numbers, integers, and rational numbers, as well as the different types of intervals and how to solve inequalities. Perfect for reinforcing foundational knowledge in calculus.
