I have three more questions I do not understand.
1. Two points (-1,1) and (3,-2) are graphed on a coordinate plane.
What is the shortest distance between the points?
A. 3
B. 4
C. 5
D. 7
I have no idea how to do this.
2. Which expression represents the nth term in the sequence 1, 3, 5, 7, 9, n...?
A. n-2
B. 2n(squared, exponent 2)-1
C. 2n-1
D. 2(exponent n)-1
I know that n is 11, but I don't get the expressions in the answers.
3. Which statement is correct?
A. All intergers are rational numbers.
B. All irrational numbers are whole numbers.
C. A real number must be a rational number.
D. A repeating decimal is an irrational number
I think it's either A or C.
1. Ah, the shortest distance between two points... It's like trying to find the shortest time to the bathroom during a fire drill. But fear not, I can help you with this! We can use the distance formula to find the answer. It goes a little something like this: square root of [(x2 - x1)^2 + (y2 - y1)^2]. In our case, the coordinates are (-1, 1) and (3, -2). Plug them into the formula, do some math magic, and you'll be golden!
2. Ah, sequences, they're like the back-to-back episodes of a sitcom. Now, the expression representing the nth term in this sequence is like the punchline, it's what ties it all together. In this case, think about what's happening to the numbers as we go from 1 to 3 to 5... Notice anything? It's increasing by 2 each time! So the expression should capture that... Can you spot the right one?
3. Rational, irrational, repeating decimals... It's a real number circus out here! Let's see if we can tame this one. So, integers are like those well-behaved kids at a fancy dinner party - they're completely rational. But irrational numbers, well, they're a bit wacky. They can't be expressed as fractions or decimals, without repeating or terminating. As for real numbers, they're quite inclusive, but being rational is not a requirement. And repeating decimals... well, it's like that annoying joke that keeps going on and on, they're irrational, my friend. So, which statement do you think is the real deal here?
1. To find the shortest distance between two points, you can use the distance formula. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, the coordinates of the two points are (-1, 1) and (3, -2). Plugging these values into the formula, we get:
d = √((3 - (-1))^2 + (-2 - 1)^2)
= √((4)^2 + (-3)^2)
= √(16 + 9)
= √25
= 5
Therefore, the shortest distance between the two points (-1, 1) and (3, -2) is 5. So the correct answer is C. 5.
2. The given sequence is 1, 3, 5, 7, 9, n...
To find the expression for the nth term in the sequence, we can observe that the sequence is an arithmetic sequence with a common difference of 2.
The formula for the nth term of an arithmetic sequence is given by:
an = a1 + (n - 1)d
In this case, the initial term a1 is 1, and the common difference d is 2. Plugging these values into the formula, we get:
an = 1 + (n - 1)2
= 1 + 2n - 2
= 2n - 1
Therefore, the expression for the nth term of the sequence is 2n - 1. So the correct answer is C. 2n - 1.
3. Let's evaluate each statement:
A. All integers are rational numbers.
This statement is true. Integers can be expressed as fractions with a denominator of 1, and any number that can be expressed as a fraction is considered rational.
B. All irrational numbers are whole numbers.
This statement is false. Irrational numbers cannot be expressed as fractions or terminating decimals, and they include numbers like √2 or π. Whole numbers, on the other hand, are non-negative integers (0, 1, 2, ...), which can be rational.
C. A real number must be a rational number.
This statement is false. Real numbers include both rational and irrational numbers. All rational numbers are real numbers, but not all real numbers are rational.
D. A repeating decimal is an irrational number.
This statement is false. Repeating decimals can be expressed as fractions, which classifies them as rational numbers. Examples include 0.333... (1/3) and 0.666... (2/3).
Therefore, the correct answer is A. All integers are rational numbers.
1. To find the shortest distance between two points on a coordinate plane, you can use the distance formula. The distance formula is derived from the Pythagorean theorem and is given by:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, the coordinates of the two points are (-1,1) and (3,-2). Plugging these values into the distance formula, we get:
Distance = √((3 - (-1))² + (-2 - 1)²)
= √(4² + (-3)²)
= √(16 + 9)
= √25
= 5
Therefore, the shortest distance between the points is 5. Therefore, the correct answer is C. 5.
2. The given sequence starts with 1 and each subsequent term increases by 2. So, the nth term can be represented as 1 + (n-1) * 2. Simplifying this expression, we get:
1 + (n-1) * 2 = 1 + 2n - 2 = 2n - 1
Therefore, the correct answer is C. 2n-1.
3. Let's go through each statement:
A. All integers are rational numbers.
This statement is correct. Integers are whole numbers (positive, negative, or zero) and can be expressed as ratios of two integers. Therefore, they are rational numbers.
B. All irrational numbers are whole numbers.
This statement is incorrect. Irrational numbers are numbers that cannot be expressed as a ratio of two integers and have non-repeating, non-terminating decimal representations. Whole numbers, on the other hand, are integers. Not all irrational numbers are whole numbers.
C. A real number must be a rational number.
This statement is incorrect. Real numbers include both rational and irrational numbers. While rational numbers can be expressed as ratios of two integers, irrational numbers cannot.
D. A repeating decimal is an irrational number.
This statement is incorrect. Repeating decimals, also known as recurring decimals, can be expressed as ratios of two integers, making them rational numbers. Irrational numbers are represented by non-repeating, non-terminating decimals.
Therefore, the correct answer is A. All integers are rational numbers.