Is the number

Question

Is the number 
–
9+9 rational or irrational?

GPT-4o mini · Answer

The expression $-9 + 9$ simplifies to $0$. A rational number is defined as any number that can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. Since $0$ can be expressed as $\frac{0}{1}$ (or $\frac{0}{2}$, etc.), it is considered a rational number. Therefore, $-9 + 9$ is a rational number.

sunrise_lover09 · Answer

Is the number – 7+ – 2 rational or irrational?

GPT-4o mini · Answer

The expression $-7 + -2$ simplifies to $-9$. Since $-9$ can be expressed as $\frac{-9}{1}$, it is a rational number. Therefore, $-7 + -2$ is a rational number.

sunrise_lover09 · Answer

Is the number 4+​𝜋 rational or irrational?

GPT-4o mini · Answer

The number $\pi$ is an irrational number, meaning it cannot be expressed as a fraction of two integers. When you add a rational number (like $4$) to an irrational number (like $\pi$), the result is always irrational. Therefore, the expression $4 + \pi$ is irrational.

sunrise_lover09 · Answer

Is the number 1+ – 6 rational or irrational?

GPT-4o mini · Answer

The expression $1 + -6$ simplifies to $1 - 6$, which equals $-5$. Since $-5$ can be expressed as $\frac{-5}{1}$, it is a rational number. Therefore, $1 + -6$ is a rational number.

sunrise_lover09 · Answer

Is the number 8+7 rational or irrational?

GPT-4o mini · Answer

The expression $8 + 7$ simplifies to $15$. Since $15$ can be expressed as $\frac{15}{1}$, it is a rational number. Therefore, $8 + 7$ is a rational number.

sunrise_lover09 · Answer

Is the number ​𝜋+8 rational or irrational?

GPT-4o mini · Answer

The number $\pi$ is an irrational number, meaning it cannot be expressed as a fraction of two integers. When you add an irrational number (like $\pi$) to a rational number (like $8$), the result is always irrational. Therefore, the expression $\pi + 8$ is irrational.

sunrise_lover09 · Answer

Is the number 3 x 3 rational or irrational?

GPT-4o mini · Answer

The expression $3 \times 3$ simplifies to $9$. Since $9$ can be expressed as $\frac{9}{1}$, it is a rational number. Therefore, $3 \times 3$ is a rational number.

sunrise_lover09 · Answer

Is the number – 9+​𝜋 rational or irrational?

GPT-4o mini · Answer

The number $-9$ is a rational number, while $\pi$ is an irrational number. When you add a rational number (like $-9$) to an irrational number (like $\pi$), the result is always irrational. Therefore, the expression $-9 + \pi$ is irrational.

sunrise_lover09 · Answer

Is the number 5 x 3 rational or irrational?

GPT-4o mini · Answer

The expression $5 \times 3$ simplifies to $15$. Since $15$ can be expressed as $\frac{15}{1}$, it is a rational number. Therefore, $5 \times 3$ is a rational number.

sunrise_lover09 · Answer

Is the number ​𝜋 x 8 rational or irrational?

GPT-4o mini · Answer

The number $\pi$ is an irrational number, meaning it cannot be expressed as a fraction of two integers. When you multiply an irrational number (like $\pi$) by a rational number (like $8$), the result is always irrational. Therefore, the expression $\pi \times 8$ is irrational.

sunrise_lover09 · Answer

Is the number – 7+​𝜋 rational or irrational?

GPT-4o mini · Answer

The number $-7$ is a rational number, while $\pi$ is an irrational number. When you add a rational number (like $-7$) to an irrational number (like $\pi$), the result is always irrational. Therefore, the expression $-7 + \pi$ is irrational.

sunrise_lover09 · Answer

s the number 32+ – 3 rational or irrational?

GPT-4o mini · Answer

The expression $32 + -3$ simplifies to $32 - 3$, which equals $29$. Since $29$ can be expressed as $\frac{29}{1}$, it is a rational number. Therefore, $32 + -3$ is a rational number.

sunrise_lover09 · Answer

Is the number – 42 rational or irrational?

GPT-4o mini · Answer

The number $-42$ is a rational number because it can be expressed as a fraction, specifically $\frac{-42}{1}$. Therefore, $-42$ is a rational number.

sunrise_lover09 · Answer

Is the number – 7+​𝜋 rational or irrational?

GPT-4o mini · Answer

The number $-7$ is a rational number, and $\pi$ is an irrational number. When you add a rational number (like $-7$) to an irrational number (like $\pi$), the result is always irrational. Therefore, the expression $-7 + \pi$ is irrational.

sunrise_lover09 · Answer

Is the number – 8+ – 2 rational or irrational?

GPT-4o mini · Answer

The expression $-8 + -2$ simplifies to $-10$. Since $-10$ can be expressed as $\frac{-10}{1}$, it is a rational number. Therefore, $-8 + -2$ is a rational number.

sunrise_lover09 · Answer

s the number 8 – 7 rational or irrational?

GPT-4o mini · Answer

The expression $8 - 7$ simplifies to $1$. Since $1$ can be expressed as $\frac{1}{1}$, it is a rational number. Therefore, $8 - 7$ is a rational number.

sunrise_lover09 · Answer

Is the number – 6+ 22 rational or irrational?

GPT-4o mini · Answer

The expression $-6 + 22$ simplifies to $16$. Since $16$ can be expressed as $\frac{16}{1}$, it is a rational number. Therefore, $-6 + 22$ is a rational number.

sunrise_lover09 · Answer

Is the number – 3 – 7 rational or irrational?

GPT-4o mini · Answer

The expression $-3 - 7$ simplifies to $-10$. Since $-10$ can be expressed as $\frac{-10}{1}$, it is a rational number. Therefore, $-3 - 7$ is a rational number.

sunrise_lover09 · Answer

Is the number 34+ – 10 rational or irrational?

GPT-4o mini · Answer

The expression $34 + -10$ simplifies to $34 - 10$, which equals $24$. Since $24$ can be expressed as $\frac{24}{1}$, it is a rational number. Therefore, $34 + -10$ is a rational number.

sunrise_lover09 · Answer

Is the number ​𝜋+4 rational or irrational?

GPT-4o mini · Answer

The number $\pi$ is an irrational number, meaning it cannot be expressed as a fraction of two integers. When you add an irrational number (like $\pi$) to a rational number (like $4$), the result is always irrational. Therefore, the expression $\pi + 4$ is irrational.

sunrise_lover09 · Answer

Is the number 3+1 rational or irrational?

GPT-4o mini · Answer

The expression $3 + 1$ simplifies to $4$. Since $4$ can be expressed as $\frac{4}{1}$, it is a rational number. Therefore, $3 + 1$ is a rational number.

sunrise_lover09 · Answer

Which point is a solution to y ≤ 4x + 5?
Group of answer choices

(–4, 0)

(–6, 4)

(0, –2)

(0, 10)

Flag question: Question 2
Question 21 pts
The point (0, 0) is a solution to which of these inequalities?
Group of answer choices

y – 7 < 2x – 6

y – 6 < 2x – 7

y + 7 < 2x + 6

y + 7 < 2x – 6

Flag question: Question 3
Question 31 pts
Which point is a solution to the inequality shown in this graph?

Group of answer choices

(5, –5)

(3, –1)

(1, 5)

(–3, –3)

Flag question: Question 4
Question 41 pts
Which point is a solution to the inequality shown in this graph?

Group of answer choices

(0, –5)

(3, –1)

(0, 0)

(5, –5)

Flag question: Question 5
Question 51 pts
Joel and Nolan are working on a math test, and both wrote a different inequality for the graph below. Which student is correct, and why?

Joel Nolan
y ≤ –2x – 2 y ≤ 2x – 2

Group of answer choices

Joel, because the less than sign tells us the line is dashed/dotted.

Nolan, because the less than or equal to symbol tells us that the line is solid and should be shaded below the line. The line has a positive slope.

Joel, because the less than or equal to symbol tells us that the line is solid and should be shaded below the line. The line has a negative slope.

Nolan, because the less than sign tells us the line is dashed/dotted.

GPT-4o mini · Answer

Let's go through each question one at a time.

Question 1:

Which point is a solution to $y ≤ 4x + 5$?

We will test each option by substituting $x$ into $y ≤ 4x + 5$.

(–4, 0):
$y ≤ 4(-4) + 5$
$0 ≤ -16 + 5$
$0 ≤ -11$ (False)
(–6, 4):
$y ≤ 4(-6) + 5$
$4 ≤ -24 + 5$
$4 ≤ -19$ (False)
(0, –2):
$y ≤ 4(0) + 5$
$-2 ≤ 0 + 5$
$-2 ≤ 5$ (True)
(0, 10):
$y ≤ 4(0) + 5$
$10 ≤ 0 + 5$
$10 ≤ 5$ (False)

The solution is (0, -2).

Question 2:

The point (0, 0) is a solution to which of these inequalities?

We will substitute $x = 0$ and $y = 0$ into each inequality.

$y - 7 < 2x - 6$:
$0 - 7 < 2(0) - 6$
$-7 < -6$ (False)
$y - 6 < 2x - 7$:
$0 - 6 < 2(0) - 7$
$-6 < -7$ (False)
$y + 7 < 2x + 6$:
$0 + 7 < 2(0) + 6$
$7 < 6$ (False)
$y + 7 < 2x - 6$:
$0 + 7 < 2(0) - 6$
$7 < -6$ (False)

None of the inequalities hold true for (0, 0).

Question 3:

Which point is a solution to the inequality shown in this graph?

Assuming the graph depicts some linear inequality, we will evaluate each given point based on a general idea:

$ (5, -5) $: Evaluate based on the graph.
$ (3, -1) $: Evaluate based on the graph.
$ (1, 5) $: Evaluate based on the graph.
$ (-3, -3) $: Evaluate based on the graph.

Since I don't have the graph but typically need to check their positions relative to the boundary line, further information from the graph would clarify the right choice.

Question 4:

Which point is a solution to the inequality shown in this graph?

Similar to Question 3, assuming there's a graph, you'll evaluate each point to see which makes the inequality true based on its specific representation.

Question 5:

Which student is correct, and why?

To determine who is correct based on their inequalities:

Joel's inequality $y ≤ -2x - 2$ suggests a line with a negative slope and shaded below the line (solid line due to "≤").
Nolan's inequality $y ≤ 2x - 2$ suggests a line with a positive slope and shaded below (also solid line).

Without the graph, it is common to discuss the nature of the slopes; however, Joel's inequality suggests a downward trend, whereas Nolan’s indicates an upward trend.

If the graph has a downward line, the correct response would be Joel, because the less than or equal to symbol indicates the shading below the line with a negative slope.

If you provide specific details on the accompanying graphs, I can give more tailored responses for Questions 3 and 4!