Write the phrase “all real numbers that are greater than −4 and less than or equal to 7” as a compound inequality.

Final answer:

To determine the number of rows in a theater that can seat 864 people, with the number of rows being 5 less than the number of seats per row, we set up the equation x(x-5)=864. Solving this quadratic equation, we find that there are 31 rows of seats.

Explanation:

To solve for the number of rows in the theater, let's denote the number of seats in each row as x and the number of rows as x - 5. Since the product of the number of seats in each row and the number of rows will give us the total seating capacity, we can write down the equation:

x(x - 5) = 864

Expanding the equation, we get:

x2 - 5x - 864 = 0

Using the quadratic formula or factoring, we find that x = 36. Therefore:

The number of rows is x - 5 = 36 - 5 = 31.

So, the theater has 31 rows of seats.

collect like terms;

3x-10x=-7x

-7x+5=5x+17

move the variable to the left and change sign

-7x+5=5x+17

move the constant to the right and change sign

-7x-5x+17-5

collect the like terms

-12x=17-5

subtract the numbers

-12x=12

divide both sides by -12

your answer is: x= -1

Answer:

y=3x

Step-by-step explanation:

For a line to be proportional it must have the form y=mx where the y-intercept is (0,0) through the origin. To change y=3x+7 to be proportional, write it as y=3x.

Answer:

The constant rate of change of the account is $50 or Increasing by $50 per month.

Step-by-step explanation:

Consider the provided information.

Joanne is depositing money into a bank account. After 3 months there is $150 in the account. After 6 months there is $300 in the account.

Rate of change is known as how one quantity change in relation to other.

The rate of change can be calculated as:

[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

Now use the above formula to calculated the rate of change.

[tex]\frac{300-150}{6-3}[/tex]

[tex]\frac{150}{3}[/tex]

[tex]50[/tex]

Hence, the constant rate of change of the account is $50 or Increasing by $50 per month.

Answer: The solution is,

[tex]n = 0\text{ or }n = \frac{1}{10}[/tex]

Step-by-step explanation:

Since, Zero product property states that if the product of two numbers or expression is equal to zero then either of the numbers or expressions must be equal to zero.

That is, If a.b = 0 ⇒ a = 0 or b = 0

Here, the given expression is,

[tex]-8n(10n-1)=0[/tex]

[tex]\implies (-8n)(10n-1)=0[/tex]

Thus, by the above property,

[tex]-8n = 0\text{ or }(10n-1)=0[/tex]

[tex]\implies n = \frac{0}{-8}\text{ or }10n= 1[/tex]

[tex]\implies n = 0\text{ or }n = \frac{1}{10}[/tex]

Answer: [tex]n^{2}[/tex]

Step-by-step explanation:

(n + n) represents the definition of the sum of n and itself.

(2n) means two times n, wich is the same as adding n + n.

[tex]n^{2}[/tex] actually represents the multiplication of n two times: [tex]n*n[/tex]

For example, if n=3:

[tex]n+n=3+3=6[/tex]

[tex]2n=2(3)=6[/tex]

[tex]n^{2}=3^{2} =9[/tex]

Answer:

f(x) = don't spend more than 200 dollars on staplers

Step-by-step explanation:

The expected value E(y) cannot be computed without the given probability density function for the random variable y. The computation involves integrating the product of the variable and its probability density over its range. An example has been provided for how to compute the expected value with a hypothetical p.d.f.

To answer the student's question, it appears there may be a missing probability density function (p.d.f.) for the random variable y, which represents the total amount of gas pumped at the station in a month. Without the actual function, we cannot compute E(y), which is the expected value (mean) of the random variable y. In general, if we had the p.d.f., we would compute the expected value by integrating the product of the variable and its probability density over its range.

An example of how to compute the expected value if we had a p.d.f. f(y) might look like this:

E(y) = ∫ y × f(y) dy

For instance, if the p.d.f. were f(y) = ay (a being a constant) over the interval [0, 1], then:

However, without the p.d.f. provided, we can only state the method, not the actual expected value.

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There are 27 possible outcomes for your favorite football team's remaining three games.

Probability is defined as the possibility of an event being equal to the ratio of the number of favorable outcomes and the total number of outcomes.

If each game can have one of three possible outcomes (win, lose, or tie), then the total number of possible outcomes for all three games can be calculated by multiplying the number of possible outcomes for each individual game.

Since each game has three possible outcomes, the total number of possible outcomes for all three games is:

3 x 3 x 3 = 27

So there are 27 possible outcomes for your favorite football team's remaining three games.

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The rate of depreciation of the piece of equipment is 27.45%.

Given that,

Based on the above information, the depreciation should be

= (Office equipment cost - scrap value) ÷ (Office equipment cost × number of years)

= ($17,000 - $3,000) ÷ ($17,000 × 3)

= $14,000 ÷ $51,000

= 27.45%

Therefore we can conclude that the rate of depreciation of the piece of equipment is 27.45%.

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Trigonometric Identities are equalities that utilize trigonometry functions and hold true for all variables in the equation. The correct option is C, -7/25.

Trigonometric Identities are equalities that utilize trigonometry functions and hold true for all variables in the equation. There are several trigonometric identities relating to the side length and angle of a triangle.

Given that the value of tan(x) = -4/3 and x is in quadrant 2.

In order to find the value of cos(2x), we can use the trigonometric identity of cos(2x), therefore, for cos(2x) we can write,

[tex]\cos(2x) = \dfrac{1-\tan^2(x)}{1+\tan^2(x)}[/tex]

Since the value of tan(x) is known, substitute the value in the identity,

[tex]\cos(2x) = \dfrac{1-(-\frac{4}{3})^2}{1+(-\frac{4}{3})^2}\\\\\cos(2x) = \dfrac{1-(\frac{16}{9})}{1+(\frac{16}{9})}\\\\[/tex]

cos(2x) = (-7/9) × (9/25)

Cancelling 9 from the numerator and the denominator,

cos(2x) = -7/25

Hence, if tanx=-4/3 and x are in quadrant 2 then cos2x=-7/25.

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The correct answer is c. [tex]$\cos(2x) = -\frac{7}{25}$[/tex].

First, we find [tex]$\sin(x)$[/tex] and [tex]$\cos(x)$[/tex] using the given value of [tex]$\tan(x)$[/tex]:

[tex]$\tan(x) = \frac{\sin(x)}{\cos(x)} = -\frac{4}{3}$.[/tex]

We can use the Pythagorean identity [tex]$\sin^2(x) + \cos^2(x) = 1$[/tex] to find [tex]$\sin(x)$[/tex] and [tex]$\cos(x)$[/tex]. Let's assume [tex]$\sin(x) = \frac{4}{5}$[/tex] (since [tex]$\tan(x) = -\frac{4}{3}$[/tex], and we are looking for a positive value of [tex]$\sin(x)$[/tex] in quadrant 2). Using the Pythagorean identity:

[tex]$\left(\frac{4}{5}\right)^2 + \cos^2(x) = 1$ \\ $\frac{16}{25} + \cos^2(x) = 1$ \\ $\cos^2(x) = 1 - \frac{16}{25}$ \\ $\cos^2(x) = \frac{25}{25} - \frac{16}{25}$ \\ $\cos^2(x) = \frac{9}{25}$. \\[/tex]

Since [tex]$\cos(x)$[/tex] is negative in quadrant 2, we have [tex]$\cos(x) = -\frac{3}{5}$[/tex].

 Now, we can use the double-angle formula for cosine:

[tex]$\cos(2x) = 2\cos^2(x) - 1$ \\ $\cos(2x) = 2\left(-\frac{3}{5}\right)^2 - 1$ \\ $\cos(2x) = 2\left(\frac{9}{25}\right) - 1$ \\ $\cos(2x) = \frac{18}{25} - 1$ \\ $\cos(2x) = \frac{18}{25} - \frac{25}{25}$ \\ $\cos(2x) = -\frac{7}{25}$.[/tex]

Therefore, the value of [tex]$\cos(2x)$[/tex] = [tex]$ -\frac{7}{25}$[/tex].

To calculate Mary's car's gas mileage, divide the number of miles driven by the number of gallons of gas used. In this scenario, her car's gas mileage is approximately 19.06 miles per gallon.

To calculate Mary's car's gas mileage, we need to divide the number of miles driven by the number of gallons of gas used. In this scenario, Mary traveled 324 miles and used 17 gallons of gas. Therefore, her car's gas mileage can be calculated as:

Gas mileage = Miles driven / Gallons of gas used

Substituting the values:

Gas mileage = 324 miles / 17 gallons

Simplifying the equation:

Gas mileage = 19.06 miles per gallon (rounded to two decimal places)

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The prime factorization of 37 is 37

We have,

The prime factorization of a number involves expressing it as a product of prime numbers.

However, in the case of prime numbers themselves, their prime factorization is simply the number itself.

In this case,

The number 37 is a prime number because it is only divisible by 1 and itself.

Since it has no other factors, its prime factorization is simply 37.

Thus,

The prime factorization of 37 is 37.

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Answer: c

Step-by-step explanation: