Simplify by reducing. Begin by determining the domain restrictions. Show all work.

x2 + 6x + 9 over x2 - 9

x2+6x+9/x2-9

3x2 - 9x over x2 - x - 6

3x2-9x/x2-x-6

Answer:

Step-by-step explanation:

The choice that best describes the number of smoothies that Savannah could buy between now and the end of this month with her gift card balance is "She can buy from 0 to 7 smoothies, but no more."

Given : 40 - 4x > 10 which is an inequality

To find : The value of x

Procedure:

Step 1:  Collect all the terms with x on right side and the constant terms  on the left side of the greater sign.

40 - 4x > 10

When 10 is brought to the left, there will be change in sign for the number 10 and similarly, when -4x is taken to the right, it becomes +4x or simply 4x. So inequality becomes

40 - 10 > 4x

Step 2: On solving the resulting inequality, we get

30 > 4x

Divide both sides by 4. So inequality becomes

[tex]\frac{30}{4} > \frac{4x}{4}[/tex]

7.5 > x

This means, x < 7.5

The choice that best describes the number of smoothies that Savannah could buy between now and the end of this month with her gift card balance is "She can buy from 0 to 7 smoothies, but no more."

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Answer:She can buy from 0 to 7 smoothies, but no more."

Step-by-step explanation:She can buy from 0 to 7 smoothies, but no more." Which is option B

The equation of the polynomial P3 is  p3​(t) = 5/6t³− 17/6t​

Equation:

In algebra, an equation consists of variable, number and constants.

Given,

Here we need to find the polynomial P3 such that {P0, P1, P2, P3} is an orthogonal basis for the subspace P3 of P4.

And the scale the polynomial p3 so that its vector of values is (-1,2,0, -2,1).

Here we know that the polynomial (before the scaling) p3​ is just the difference between t³ and its orthogonal projection on the span of 1,t,t²−2.

Then we showed that the projection is ​, hence

p3​(t) = t³− 17/5t​

When here we have the  scale this polynomial such that the vector of values at t=−2,−1,0,1,2 is (−1,2,0,−2,1).

Now, we need to find the scalar α, it can be written as,

=> α⋅p3​(−2)=−1

=> α⋅p3​(−1)=2

=> α⋅p3​(0)=0

=> α⋅p3​(1)=−2

=> α⋅p3​(2)=1

Here from the 4-th equation we have the value of

=> α⋅(1- 17/5) = -2

=> α = 5/6​

And it is easy to check that all the equations are satisfied with this α.

Therefore, the required polynomial p3 is

=> p3​(t) = 5/6(t³− 17/5t​)

=> p3​(t) = 5/6t³− 17/6t​

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Answer:

  (7, -2)

Step-by-step explanation:

You want a graphical solution to the equations ...

The graph is shown in the attachment. The solution is (x, y) = (7, -2).

He invested in each account as follows:

First account ----->  31,000

Second account ----->  10,000

What is the percentage?

It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.

gained - lost = 1670

0.07x - 0.05(41000 - x) = 1670

Find x :

0.07x - 0.05(41000 - x) = 1670

0.07x - 2050 + 0.05x = 1670

0.12x = 1670  + 2050

x = 31,000

41000 - 31,000 = 10,000

He invested in each account as follows:

First account ----->  31,000

Second account ----->  10,000

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Ask a question action correctly completes this graphic organizer.

Option A is correct .

A graphic organizer is a visual and graphic display that depicts the relationships between facts, terms, and or ideas within a learning task.Graphic organizers are also sometimes referred to as knowledge maps, concept maps, story maps, cognitive organizers, advance organizers, or concept diagrams

.Ask a question action correctly completes this graphic organizer.

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The values to make up the solution set of inequality are {8,9,10}.

In mathematics, an unjust comparison between two numbers or other mathematical expressions is referred to as an inequality.

A less than symbol (a, b) denotes that one thing is less than the other.

The notation a > b indicates that an is greater than b.

A and b are not equal in either situation. A strict inequality is one in which an is strictly less than or strictly bigger than b in certain connections. Comparability is left out.

First we have to replace m with 8, it becomes:

8+7 =15<18

For 9,  

9+7 =16<18

For 10,

10+7 =17<18

And for 11,

11+7 =18=18

So, The values to make up the solution set of inequality are {8,9,10}.

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a) Corresponding angles are congruent, m<a=139

b) Alternate Interior Angles are congruent, m<b =139

The linear regression for the given data is y = 0.8x + 12.78 and the test grade will be 44.

What is linear regression?

A variable's value can be predicted using linear regression analysis based on the value of another variable. The dependent variable is the one you want to be able to forecast. The independent variable is the one you're using to make a prediction about the value of the other variable.

Based on the given data

Using the graphing calculator, the linear regression will be,

y = 0.8x + 12.78

For a homework grade of 39, x = 39

Then, y = 0.8*39 + 12.78 = 43.98 ≈ 44

Hence, the test grade will be 44.

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Answer:

Step-by-step explanation:

The vertex of p(x)  is (9, 0).

Vertex of q(x) is (0, 9).

Vertex of r(x) is (-9, 0).

Vertex of t(x) is (0, -9).

What is absolute value function?

The non-negative value of a real number x without regard to its sign is known as its absolute value or modulus in mathematics and is denoted by the symbol |x|. Specifically, |x|=x and |x|=-x, respectively, if x is a positive number, and |0|=0 otherwise.

The vertex form of an absolute value function is f(x) = a |x - h| + k where

(h, k) is the vertex from which the graph starts and a = m is the slope.

So, the absolute value function  p(x) = |x - 9|, means h =-(-9) = 9 and

k = 0, will result in the vertex at (9, 0).

The absolute value function q(x) = |x| +9 , means h = 0 h=0 and k =9, will result in a vertex (0, 9).

The absolute value function r(x) = |x + 9|,  means, h = -9 and, k=0, will result in the vertex at (-9, 0).

The absolute value function t(x) = |x| -9 , means h = 0 and k = -9, will result in the vertex at (0, -9)

Hence,

The vertex of p(x)  is (9, 0).

Vertex of q(x) is (0, 9).

Vertex of r(x) is (-9, 0).

Vertex of t(x) is (0, -9).

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Answer:2

Step-by-step explanation:

Combine like terms and divide

8x/4x

2

a. The population of the state in 2000 is 18.5 million

b. The population will reach 26.6 million in about 2 years

From the question, we are to determine the population of the state in 2000.

From the given information,

The formula that models the population of the US state after 2000 is

A = 18.5e^(0.1708t)

To determine the population of the state in 2000, we will substitute t = 0

That is,

A = 18.5e^(0.1708(0))

A = 18.5e^(0)

A = 18.5(1)

A = 18.5

Thus, the population of the state in 2000 is 18.5 million

b. To determine when the population will reach 26.6 million

Substitute A = 26.6 in the equation

A = 18.5e^(0.1708t)

26.6 = 18.5e^(0.1708t)

Solve for t

26.6/18.5 = e^(0.1708t)

ln(26.6/18.5) = 0.1708t

0.36314048 = 0.1708t

t = 0.36314048/0.1708

t = 2.126

t ≈ 2

Hence, the population will reach 26.6 million in about 2 years

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The answer to this Question based on pH is 8.12

What is pH?

pH is a measure of acidity or basicity in a solution. It is measured on a scale ranging from 0 to 14, with 7 being neutral. A pH below 7 is acidic and a pH above 7 is basic. A solution with a pH of 0 is the most acidic, while a solution with a pH of 14 is the most basic. The pH of a solution is an important factor in many chemical and biological processes, as it affects the activity of molecules. For example, enzymes typically only function within a certain pH range, and most biological cells have a preference for a specific pH. pH is also important in determining the solubility of certain substances, and can affect the taste of food.

Here [H] = 7.7 * 10^-9

pH = -log[7.7 * 10^-9]

using some log properties this value comes out to be

pH = 9 - 0.88

pH = 8.12 and pH greater than 7 means solution is Basic

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The solution set for absolute value inequalities are presented as follows;

(b) (i) -∞ < x ≤ 2

(ii) 1.25 ≤ x < 3, or x > 3

An absolute value inequality is an inequality with an absolute value expression containing a variable.

(i) |2·x + 7| + 1 ≥ 6·x

Solving for when |2·x + 7| is positive, we get;

2·x + 7 + 1 ≥ 6·x

7 + 1 ≥ 6·x - 2·x = 4·x

8 ≥ 4·x

Dividing both sides by 4, we get;

8 ÷ 4 ≥ 4·x ÷ 4 = x

2 ≥ x

Therefore; x ≤ 2

The solution of the absolute value inequality can be found as follows;

|2·x + 7| + 1 ≥ 6·x

|2·x + 7| ≥ 6·x - 1

Therefore, we have the following compound inequality;

2·x + 7 ≤ -(6·x - 1), 2·x + 7 ≥ (6·x - 1)

The solution for the inequality, 2·x + 7 ≤ -(6·x - 1) is found as follows;

2·x + 7 ≤ -(6·x - 1) = 1 - 6·x

2·x + 6·x ≤ 1 - 7 = 6

8·x ≤ 6

x ≤ 6/8 = 3/4

x ≤ 3/4

The solution for the inequality, 2·x + 7 ≥ (6·x - 1) is found as follows;

2·x + 7 ≥ (6·x - 1)

7 + 1 ≥ 6·x - 2·x = 4·x

8 ≥ 4·x

x ≤ 2

Combining the solution, we get;

-∞ < x ≤ 2

(ii) [tex]\left|\dfrac{2\cdot x+ 1}{x - 3} \right| \geq 2[/tex]

Therefore, we get;

[tex]\dfrac{2\cdot x+ 1}{x - 3} \leq -2[/tex]

[tex]\dfrac{2\cdot x+ 1}{x - 3} + 2 \leq 0[/tex]

[tex]\dfrac{4\cdot x - 5}{x - 3} \leq 0[/tex]

x < 3, or x ≥ 1.25

1.25 ≤ x < 3

[tex]\dfrac{2\cdot x+ 1}{x - 3} \geq 2[/tex]

[tex]\dfrac{2\cdot x+ 1}{x - 3} -2 \geq 0[/tex]

[tex]\dfrac{7}{x - 3} \geq 0[/tex]

Therefore, x > 3,

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Answer: 8 Five point questions

Step-by-step explanation:

Hope this helps you!

(a) As time goes on and approaches infinity, the level of oxygen in the pond approaches normal, therefore;

The oxygen level will eventually approach its  normal level in the long-run

(b)  The number of weeks after which the oxygen level becomes 60% of its normal level are 0.5, 2 weeks

The rational polynomial function  in which the numerator and denominator have the same power and the coefficient of the highest power are the same indicates that the value of f(t) approaches 1 (normal) as t approaches infinity.

The function with which the oxygen level in the pond can be found is presented as follows;

[tex]f(t) = \dfrac{t^2-t + 1}{t^2+ 1}[/tex]

f(0) = 1, therefore

The end behavior of the function as the value ot t approaches infinity is therefore the asymptote;

y = 1/1 = 1

The  value of the function f(t) as the value of t approaches infinity is that the f(t) approaches 1, which is the normal level of oxygen in the pond

(b) When the oxygen level is 60%, we get;

[tex]f(t) = \dfrac{t^2-t + 1}{t^2+ 1} = 60\% = 0.6[/tex]

0.6·(t² + 1) = t² - t + 1

0.6·t² + 0.6 = t² - t + 1

0.4·t² - t + 0.4 = 0

t² - 2.5·t + 1 = 0

Therefore;

(t - 2)·(t - 0.5) = 0

t = 2 or t = 0.5

The times when the oxygen level is 60% is 0.5 and 2 weeks after the organic waste is dumped in the pond.

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If you only need 1 meter and 45 cm. Then the length of the board left will be 1.55 meters.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

Conversion means converting the same thing into different units.

Home Depot sells boards in 3-meter lengths. If you only need 1 meter and 45 cm. Then the length that you need is calculated as,

⇒ 1 meter and 45 centimeters

Convert the centimeters into a meter. Then we have

⇒ 1 + 45 / 100

⇒ 1 + 0.45

⇒ 1.45

Then the length of the board left will be calculated as,

⇒ 3 - 1.45

⇒ 1.55 meters

If you only need 1 meter and 45 cm. Then the length of the board left will be 1.55 meters.

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The number of races that were 10-kilometer long is 102.

An athlete ran a total of 1,500 kilometers in a race before retiring. The athlete completed 32 races that were at least 15 kilometers in length. The rest were 10-kilometer races. We need to find out the number of races that were 10-kilometer long.

Let the number of races that were 10-kilometers long be denoted by the variable "x". An equation is a formula in mathematics that expresses the equivalence of two expressions by linking them with the equal sign. We can write the equation as given below.

32×15 + 10x = 1,500

480 + 10x = 1,500

10x = 1,500 - 480

10x = 1,020

x = 1,020/10

x = 102

Hence, the number of races that were 10-kilometer long is 102.

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The adults less than 10 minutes are 52, the teenager less than 10 minutes 72, the teenager more than 10 minutes are 64, and the total is 250.

What is mathematical operations in reasoning?

The study is based on the mathematical operations that are multiplication, addition, subtraction, and division. “Mathematical Operations reasoning” is the simple process of the expression of the containing numbers and various operations in the mathematical field.

                                              Time to complete Maze

                         less than 10 minutes         10 minutes or more          Total

Adults                           52                                      62                             114

Teenager                     72                                       64                            136

Total                            124                                      126                           250

114 - 62 = 52 {adults less than 10 minutes}

52 + 20 = 72 {teenager less than 10 minutes}

136 - 72 = 64 {teenager more than 10 minutes}

124 + 126 = 250 {total}

Hence, the adults less than 10 minutes are 52, the teenager less than 10 minutes 72, the teenager more than 10 minutes are 64, and the total is 250.

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Answer:  AB || DC   (1st choice)

Reason:

We have two pairs of consecutive interior angles that are supplementary. We then use the converse of the consecutive interior angles theorem to conclude side AB is parallel to DC. This means ABCD is a trapezoid.

Refer to the diagram below.

Answer:

y = 14

Step-by-step explanation:

y = k[tex]\sqrt{x}[/tex]  Solve for k

7 = k[tex]\sqrt{9}[/tex]

7 = k3  Divide both sides by 3 to solve for k

[tex]\frac{7}{3}[/tex] = k  

y = kx

y  = [tex]\sqr\frac{7}{3} }[/tex][tex]\sqrt{36}[/tex]

y = [tex]\frac{7}{3}[/tex](6)

y = 14

[tex]\frac{7}{3}[/tex][tex](\frac{6}{1})[/tex] is the same thing as [tex]\frac{7}{1}[/tex][tex](\frac{2}{1})[/tex]  I cross canceled the 3 and the 6

[tex]\frac{14}{1}[/tex] = 14

x = 12, using the concept of absolute value equation.

The non-negative value of a real number x, regardless of its sign, is its absolute value (or modulus), | x |. For instance, 5 has an absolute value of 5, and 5 has a value of 5. One way to think about a number's absolute value is as its distance from zero on the real number line.

To answer an absolute value problem, isolate the absolute value on one side of the equation. Then, resolve both equations by setting their respective contents to the positive and negative values of the integer on the other side of the equation. 

Given that,

4|x+6| = 24

either |x+6| = 6 or,  |x+6| = -6

x = 0. -12

So, x = 12                           [here as negative sign is already present]

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The quadratic equation is x² - 2x - 24

What is quadratic equation ?

Any equation in algebra that can be written in standard form as where x stands for an unknown value and a, b, and c stand for known values is said to be a quadratic equation. In general, it is assumed that a > 0; equations with a = 0 are regarded as degenerate since they become linear or even simpler.

Square and quadrangle difficulties have a close relationship with quadratic equations (another name for rectangles). In actuality, the Latin word quadratus, which means square, is the root of the word quadratic.

Any quadratic problem can be solved using the quadratic formula. The equation is first changed to have the form ax2+bx+c=0, where a, b, and c are coefficients. After that, we enter these coefficients into the following formula: (-b(b2-4ac))/(2a).

Given roots of quadratic equation are 6 and -4.

Also given here leading coefficient is 1.

We know that, the formulae for forming quadratic equation when its roots are given is as follows: x² - (α+β)x + αβ

Here α and β are the roots of the quadratic equation.

Let  α = 6 and β = -4

Putting the value of  in the above formulae we get,

x² - (6-4)x - 24

x² - 2x - 24

But remember here 1 is the leading coefficient of the quadratic equation with roots 6 and -4 (given in question),so we have to multiply the equation by 1 to get the final answer.

So, Hence the required quadratic equation is x² - 2x - 24

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18.5 is correct for part (a) since t = 0 leads to A = 18.5

=====================================================

Part (b)

Your teacher is asking for the value of t when A = 26.6

A = 18.5*e^(0.1708t)

26.6 = 18.5*e^(0.1708t)

26.6/18.5 = e^(0.1708t)

1.437838 = e^(0.1708t)

Ln(1.437838) = 0.1708t

t = Ln(1.437838)/0.1708

t = 2.126116 approximately

t = 2 when rounding down to the nearest integer

Therefore, 2 years after the year 2000, the year 2002, is when the population reaches roughly 26.6 million. The actual population will be slightly less than 26.6 million, but it's close enough.

Note that:

which helps confirm the correct value of t is between t = 2 and t = 3.

The function that describes the exponential growth is P(t) = 21100 [tex]e^{0.09t}[/tex] and the population in 2008/ will be 43348.54.

In mathematics, an exponential function is a relationship of the type y = ax, where x is an independent variable that spans the entire real number line and is expressed as the exponent of a positive number.

(a)

As per the given,

Growth rate r = 9 % = 0.09

Population in 2000 (t = 0)  is 21100

The formula for exponential growth is given as,

P(t) = P₀e^(rt)

P(t) = 21100 e^(0.09 x t)

P(t) = 21100 [tex]e^{0.09t}[/tex]

(b)

The number of population in 2008 (t = 8) will be as.,

P(8) = 21100 e^(0.09 x 8)

P(8) = 43348.54

Hence "The function that describes the exponential growth is P(t) = 21100 [tex]e^{0.09t}[/tex] and the population in 2008/ will be 43348.54".

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Answer:

A: "The theoretical probability is higher."

Step-by-step explanation:

The theoretical probability of rolling an odd number on a die is 1/2, or 0.5. Since Tami rolled the die 20 times and it landed on an odd number 6 times, the experimental probability of rolling an odd number is 6/20, or 0.3. Since 0.3 is less than 0.5, the theoretical probability is higher. Therefore, the correct answer is option A: "The theoretical probability is higher."

Answer:

D

Step-by-step explanation:

if x = 1

P(x) = 1/10

if x = 2

P(x) = 2/10

if x = 3

P(x) = 3/10

if x = 4

P(x) = 4/10

The lawyer's net worth is $307,331.

Assets are those things the organization or the individual possess. It is a source of income.

Liabilities are those items that are debt or has to pay money for it.

From the list, the assets of the lawyer are:

Car value = $41,450

Savings account balance = $13,547

Season baseball tickets = $19,400

Checking account balance = $16,309

Home Value = $271,345

Total assets = $41,450 + $13,547 + $19,400 + $16,309 + $271,345

                    = $362,051

The liabilities of the lawyer are:

Car loan = $22,890

Student loans = $12,410

Credit card balance = $19,420

Total liabilities = $22,890 + $12,410 + $19,420

                        = $54,720

Net worth = Total assets - Total liabilities

                 = $362,051 - $54,720

                 = $307,331

Hence, the net worth of the lawyer with these assets and liabilities is $307,331.

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The amount invested at the rate of 8% is; $3000

We are told that Walt made an extra $7000 last year from a part-time job. He invested part of the money at 7% and the rest at 8%.

If the amount invested is denoted as X , then we have;

0.07X + 0.08(7000 - X)

Now, since he made a total of $530, the we can equate that equation to $530 to get;

0.07X + 0.08(7000 - X) = 530

Expanding this equation gives us;

0.07X + 560 - 0.08X = 530

0.01X = 560 - 530

0.01X = 30

X = 30/0.01

X = $3000

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