Lauren is working two summer jobs, making $24 per hour tutoring and making $8
per hour clearing tables. In a given week, she can work no more than 13 total hours
and must earn no less than $160. Also, she must work no more than 8 hours tutoring
and a maximum of 5 hours clearing tables. If a represents the number of hours
tutoring and y represents the number of hours clearing tables, write and solve a
system of inequalities graphically and determine one possible solution.

The quadratic function has its solution to be x = 4/9 and x = -4/9

From the question, we have the following parameters that can be used in our computation:

f(x) = 81x^2 - 16

Express the expression as difference of two squares

So, we have the following representation

f(x) = (9x)^2 - 4^2

Apply the difference of two squares rule

So we have

f(x) = (9x - 4)(9x + 4)

This gives

(9x - 4)(9x + 4) = 0

So, we have

9x = 4 and 9x = -4

Evaluate

x = 4/9 and x = -4/9

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The number of elements in the complement of the union set of A and B is given as follows:

12.

The universal set is given as follows:

S = {1, 2, 3, ... , 23, 24, 25}.

The union between two sets is composed by the elements that belong to at least one of the sets, hence the union of sets A and B is given as follows:

A U B = {2, 3, 4, 6, 8, 12, 13, 14, 16, 18, 22, 24, 25}.

The number of elements in the union set is of:

13.

The complement of a set is composed by all the elements that belong to the universal set but to not belong to the set. As the union of A and B has 13 elements, and the universal set has 25 elements, the number of elements of the complement is given as follows:

25 - 13 = 12.

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Answer:

Step-by-step explanation:

[tex]\frac{x+6}{x^2(x^2+2)} =\frac{A}{x} +\frac{B}{x^2} +\frac{Cx+D}{x^2+2} \\x+6=Ax(x^2+2)+B(x^2+2)+(Cx+D)x^2\\equating~co-efficients ~of~same~powers~of~x\\0=A+C (of~x^3)\\0=B+D (of~x^2)\\1=2A (of~x)\\A=1/2\\6=2B (constant~term)\\B=6/2=3\\C=-A=-1/2\\D=-B=-3\\\frac{x+6}{x^2(x^2+2)} =\frac{1}{2x} +\frac{-3}{x^2} +\frac{-\frac{1}{2} x-3}{x^2+2}[/tex]

The values of x and y are x = 6 and y = 3√5

Triangle 1

From the question, we have the following parameters that can be used in our computation:

The right triangle where one of the angles is 45 degree

This is a special triangle such that

Hypotenuse = Legs * √2

So, we have

x = 3√2 * √2

Evaluate

x = 6

Triangle 2

Here, we have:

The right triangle where one of the angles is 60 degrees

So, we have

cos(60) = y/6√5

This gives

y = 6√5 * cos(60)

Evaluate

y = 3√5

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The intercepts of the parabola are A ( 6 , 0 ) and B ( -2 , 0 ) and the y intercept of the parabola is ( 0 , -12 )

A Parabola, open curve, a conic section produced by the intersection of a right circular cone and a plane parallel to an element of the cone. A parabola is a plane curve generated by a point moving so that its distance from a fixed point is equal to its distance from a fixed line

The equation of the parabola is given by

( x - h )² = 4p ( y - k )

y = a ( x - h )² + k

where ( h , k ) is the vertex and ( h , k + p ) is the focus

y is the directrix and y = k – p

The equation of the parabola is also given by the equation

y = ax² + bx + c

where a , b , and c are the three coefficients and the parabola is uniquely identified

Given data ,

Let the equation of the parabola be represented as A

Now , the value of A is

y = x² - 4x - 12   be equation (1)

On simplifying , we get

when y = 0

x² - 4x - 12 = 0

On factorizing the quadratic equation , we get

x² - 6x + 2x - 12 = 0

( x + 2 ) ( x - 6 ) = 0

So , the two values of x are

when ( x + 2 ) = 0

x = -2

And , when ( x - 6 ) = 0

x = 6

So , the x intercepts of the parabola are A ( 6 , 0 ) and B ( -2 , 0 )

The y intercept is when x = 0

So , y = 0 - 0 - 12

y = -12

And , the y intercepts of the parabola are ( 0 , -12 )

Hence , the intercepts of the parabola are solved

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The time is about 122 years and 9 months after which there will be 15,000 dollars in the account.

Mathematical functions with exponents include exponential functions. f(x) = bˣ, where b > 0 and b 1, is a fundamental exponential function.

Given:

An expression,

1[tex](e)^{0.034t}[/tex] model the balance, in thousands of dollars, where t is time.

(a).

0.034 represents the rate at which the function is expressed.

So,

0.034t = log(15000)

          t = log(15000)/0.034

          t =  122.826213502 in years.

          t = 122 years, 9 months.

Therefore,  t = 122 years, 9 months.

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4.3cm is the measure of the unknown side

According to the theorem, the product of the measurements of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment if two secant segments are drawn to a circle from an exterior point.

Applying the rule to the question;

3(y+3)= 2(2+9)

3y + 9 = 22

Subtract 9 from  both sides

3y = 22- 9

3y = 13

y = 13/3

y = 4.3cm

Hence the measure of y to the nearest tenth is 4.3cm

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Answer:

Step-by-step explanation:

similar - AA

The ΔBES is similar to ΔGES by AA similarity postulate.

Similar triangles are those triangles, where the angles of the triangles are equal and the sides are proportional.

Given a bigger triangle BEG.

There are two smaller right angled triangles BES and GES.

For ΔBES, ∠BES = 40° and ∠BSE = 90°

For ΔGES, ∠GES = 40° and ∠GSE = 90°

By AA similarity postulate, if two angles of one triangle is equal to two angles of another triangle, then the triangles are similar.

Here, two angles of ΔBES is equal to two angles of ΔGES.

So two triangles are similar.

Hence the two triangles are similar by AA similarity postulate.

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Your question is incomplete. The complete question with the image of the triangle is as given below.

The domain are the possible input while the range are the possible output of a function.

(a) The domain = [-√2, √2], the range = [0, 2]

(b) The domain = [-1, 1], the range = [0, 1]

(c) The domain = [-1, 1], the range = [0, -1]

(d) The domain = [0, 2], the range = [0, 1]

(e) The domain = [-(2 + √2), (√2 - 2)], the range = [0, 2]

Reasons:

The given functions can be expressed by the equation; (-x + 1)·(x + 1) = -x² + 1

Therefore, we have;

(a) y = f(x) + 1 = -x² + 1 + 1 = -x² + 2

The x-intercept of the above function are, x = √2, and x = -√2

Which gives;

The domain = [-√2, √2]

The range = [0, 2]

(b) y = 3·f(x) = 3 × (-x² + 1) = -3·x² + 3

At the x–intercepts, we have;

-3·x² + 3 = 0

x = ±1

The domain = [-1, 1]

The maximum value of y is given at x = 0, therefore;

= -3 × 0² + 3 = 3

The range = [0, 1]

(c) y = -f(x) = -(-x² + 1) = x² - 1

At the x–intercepts, x² - 1 = 0

x = ± 1

The domain = [-1, 1]

The minimum value of y is given at x = 0, which is y = -1

The range = [0, -1]

(d) y = f(x - 1) = -(x - 1)² + 1 = -x² + 2·x

At the x–intercepts, we have;  -x² + 2·x = 0, which gives;

(-x + 2)·x = 0

Which gives, x = 0, or x = 2

The domain = [0, 2]

The maximum value of y is given when x = -b/(2·a) = -2/(2×(-1)) = 1

y = f(1) =  -1² + 2×1 = 1

Therefore;

The range = [0, 1]

(e) y = f(x + 2) + 1 = (-(x + 2)² + 1) + 1 = -x² - 4·x - 2

At the x–intercepts, we have;  -x² - 4·x - 2 = 0, which gives;

x = -(2 + √2) or x = x = √2 - 2

The domain = [-(2 + √2), (√2 - 2)]

The maximum value of y is given when x = -4/(2)) = -2

Which gives;

-(-2)² - 4·(-2) - 2 = 2

The range = [0, 2]

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Answer:

Step-by-step explanation:

If a ball plus a bat costs 1.10 dollars and the price of the bat is 1 dollar more than the ball, what is the cost of the bat?

If we represent the price of the ball as "p", then the price of the bat will be "p + 1", since the problem tells us that the price of the bat is 1 dollar more than the price of the ball.

We also know that the sum of the price of the ball and the bat is 1.10 dollars. We can write this as an equation:

p + (p + 1) = 1.10

Simplifying and solving for p, we get:

2p + 1 = 1.10

2p = 0.10

p = 0.05

Therefore, the price of the ball is 0.05 dollars. To calculate the price of the bat, we can add 1 dollar to the price of the ball:

p + 1 = 0.05 + 1 = 1.05

So, the price of the bat is 1.05 dollars.

The number that should be added to complete the square of the expression 4x² - 8x is 1.

What is an expression?

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

To complete the square of the expression 4x² - 8x, add a constant term that will make it a perfect square trinomial.

First, factor out the coefficient of the x² term -

4x² - 8x = 4(x² - 2x)

Next, add a constant term that will make the expression inside the parentheses a perfect square.

To do this, take half of the coefficient of the x term and square it.

Half of -2 is -1, and (-1)^2 is 1.

So add 1 to the expression inside the parentheses to make it a perfect square -

4x² - 8x + 1 = 4(x² - 2x + 1)

Simplify this expression by factoring the trinomial inside the parentheses -

4x² - 8x + 1 = 4(x - 1)²

Therefore, the number is 1.

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Answer:

Step-by-step explanation:

For the 3-year loan at 6% simple interest:

Interest = Principal x Rate x Time

Interest = 12,000 x 0.06 x 3

Interest = $2,160

For the 5-year loan at 7.5% simple interest:

Interest = Principal x Rate x Time

Interest = 12,000 x 0.075 x 5

Interest = $4,500

The difference in the amount of interest Hannah would have to pay on these two loans is:

$4,500 - $2,160 = $2,340

So the answer is (B) $2,340.

Answer:3x+5x

Step-by-step explanation:

3x÷5 +x

3x+5x(multiply each term by 5)

The volume of each tile is 134 cm³.

The volume of a cuboid is the amount of space it occupies in three-dimensional space. A cuboid is a three-dimensional geometric shape with six rectangular faces, where each face has perpendicular edges to two other adjacent faces. The formula for calculating the volume of a cuboid is V = l x w x h, where V is the volume, l is the length of the cuboid, w is the width of the cuboid, and h is the height of the cuboid.

To calculate the volume of a cuboid, first measure its length, width, and height. Then, multiply the length by the width and the height, and then multiply the result by the height again. This gives the volume of the cuboid in cubic units, such as cubic inches, cubic centimeters, or cubic meters.

For example, if a cuboid has a length of 5 cm, a width of 3 cm, and a height of 2 cm, the volume can be calculated as follows:

V = l x w x h

V = 5 cm x 3 cm x 2 cm

V = 30 cubic centimeters

Therefore, the volume of the cuboid is 30 cubic centimeters.

The volume of each tile can be calculated by multiplying its length, width, and height:

Volume = 10 cm × 20 cm × 0.67 cm = 134 cm³

Therefore, the volume of each tile is 134 cm³.

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Answer:

The sum using summation notation can be written as: Σn = 2 + (-10) + (50) + (-250) + (1250).

A function that decreases by 12% every time x increases by 1 is,

y = 100(1 -0.12)ˣ.

An exponential function's curve is created by a pattern of data called exponential decay, which exhibits higher decreases over time.

The exponential decay function:

Aₙ = A₀(1-r)ˣ, where y = Final amount, A₀ = Initial amount, r = Rate of decay in decimal form, x = Time.

An exponential decay function is represented by the following equation,

y = a(1 -r)ˣ.

Here, a = 100.

And the function decreases by 12%.

So,

100 - 12 = 88

In decimals, 88 = 0.88.

So, y = 100(1 -0.12)ˣ.

Therefore, the function is y = 100(1 -0.12)ˣ.

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The expression to obtain the final probability as a reduced fraction.

A fraction is a mathematical representation of a part of a whole or a part of a group. It consists of two numbers separated by a horizontal line, where the number on the top is called the numerator, and the number at the bottom is called the denominator. The numerator represents how many parts are being considered, and the denominator represents the total number of equal parts that make up the whole or group. Fractions can be written in various forms, such as proper fractions, improper fractions, and mixed numbers, and can be added, subtracted, multiplied, and divided using specific rules and techniques.

We cannot provide a specific numerical answer without more information on the gender and grade distribution of the students. However, we can use the following formula to calculate the probability that a student is either male or earned grade A:

P (male or grade A) = P(male) + P (grade A) - P (male and grade A)

The probability of selecting a male student and the probability of selecting a student who earned a grade A are two independent events, so we can add their probabilities. However, we need to subtract the probability of selecting a male student who also earned a grade A, since we don't want to count that student twice.

If we have the probabilities of each event, we can plug them into the formula and simplify the expression to obtain the final probability as a reduced fraction.

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The mean square within treatments (MSE) is 5 when in  a completely randomized experimental design involving six treatments, 11 observations were recorded for each of the six treatments.

Mean is a measure of central tendency which represents the average value of a set of numbers. It is calculated by adding up all the values in the set and then dividing the sum by the total number of values in the set.

Here,

To find the mean square within treatments (MSE), we need to use the formula:

MSE = SSE / df

where SSE is the sum of squares within treatments and df is the degrees of freedom within treatments.

Since we are given SSTR and SST, we can find SSE using the formula:

SSE = SST - SSTR

Substituting the given values, we get:

SSE = 700 - 400 = 300

The total number of observations is:

n = 6 treatments × 11 observations/treatment = 66 observations

The degrees of freedom within treatments is:

df = n - number of treatments = 66 - 6 = 60

Therefore, the mean square within treatments is:

MSE = SSE / df

= 300 / 60

= 5

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The requried, percentage of students who study less than 2.50 hours a day is approximately 25.14 percent.

A Z-score is stated as the fractional model of data point to the mean using standard deviations.

We need to convert 2.50 hours to a standard score (z-score) using the formula,

z = (x - μ) / σ

where x is the value we want to convert, μ is the mean, and σ is the standard deviation.

z = (2.50 - 3.00) / 0.75

z = -0.67

This means that 2.50 hours is 0.67 standard deviations below the mean.

We can use a standard normal distribution table or a calculator to find the percentage of the distribution that falls below this standard score. For example, using a standard normal distribution table, we can look up the area to the left of z = -0.67 and get:

P(z < -0.67) = 0.2514

So the correct answer is (a) 25.14 percent.

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The values of the coordinates x and y is P ( 6 , 3 )

The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept

And y - y₁ = m ( x - x₁ )

y = y-coordinate of second point

y₁ = y-coordinate of point one

m = slope

x = x-coordinate of second point

x₁ = x-coordinate of point one

The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Given data ,

Let the equation of line be represented as A

Now , the value of A is

Let the first point be P ( x , y )

Let the second point be Q ( 0 , 0 )

Now , the slope of the line is m₁ = ( y₂ - y₁ ) / ( x₂ - x₁ )

Substituting the values in the equation , we get

Slope m₁ = y / x

m₁ = 1/2

So , y/x = 1/2

And , x = 2y

Now , the third point is R ( 7 , 5 )

m₂ = ( 5 - y ) / ( 7 - x )

m₂ = 2

On simplifying , we get

( 5 - y ) / ( 7 - x ) = 2

Multiply by ( 7 - x ) , we get

5 - y = 14 - 2x

Adding 2x and subtracting 5 on both sides , we get

2x - y = 9

Substitute the value of x from m₁ , we get

2 ( 2y ) - y = 0

3y = 9

Divide by 3 on both sides , we get

y = 3

And , the value of x = 6

Hence , the coordinate of the point P is P ( 6 , 3 )

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The proof is incorrect and step number 3 is the first unjustified step due to missing prior step.

A quadrilateral with equal angles and parallel opposite sides is referred to as a rectangle. Around us, there are a lot of rectangle items. The length and width of any rectangle form serve as its two distinguishing attributes. The width and length of a rectangle, respectively, are its longer and shorter sides.

The step 3 in the given explanation is unjustified step. The step needs to be place at the last in order to take in consideration all the steps and properties of a rectangle.

Hence, the proof is incorrect and step number 3 is the first unjustified step due to missing prior step.

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The requried expression is correct for any value of x. We need to check if the expression is true for all possible values of x to know if it is always correct.

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
We can simplify both sides of the equation to see if they are equivalent:

7 − 2(3 − 8x) = 7 − 2(−5x)

First, simplify the expression in the parentheses on the left side:

7 − 2(3 − 8x) = 7 − 6 + 16x

7 − 2(3 − 8x) = 16x + 1

7 - 6 + 16x = 16x + 1
1 + 16x = 16x + 1

So, the expression is correct for any value of x. However, we would need to check if the expression is true for all possible values of x to know if it is always correct.

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The dividend is 3x³ + 0x² - 1x - 1, which is option A.

In algebra, a polynomial is an expression consisting of variables (also known as indeterminates) and coefficients, which are combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example [tex]2x^2 - 3x + 1[/tex] is a polynomial of degree 2 with three terms.

The term "polynomial" comes from the Greek words "poly" (meaning "many") and "nomial" (meaning "term"). Polynomials are used in many areas of mathematics and science, including calculus, physics, and engineering.

Polynomials are classified by their degree, which is the highest exponent in the polynomial. For example, the polynomial [tex]2x^3 + 3x^2 - x + 5[/tex] is a cubic polynomial, because it has a degree of 3. Polynomials can also be classified by the number of terms they have. For example, a polynomial with only one term is called a monomial, while a polynomial with two terms is called a binomial.

Polynomials have many important properties, such as the fact that the sum and product of two polynomials is also a polynomial. Polynomials can also be factored into simpler polynomials, which can be useful in solving equations and finding roots.

Overall, polynomials are a fundamental concept in algebra and play an important role in many areas of mathematics and science.

The dividend is the product of the divisor and the quotient plus the remainder. In this problem, the divisor is -3x+7 and the quotient is[tex]3x^2-7x+9[/tex]. Using polynomial long division, we get:

[tex]3x^2 - 4x + 2[/tex]

[tex]-3x + 7 | 3x^3 + 0x^2 - 1x - 1[/tex]

[tex]-3x^3 + 7x^2[/tex]

[tex]-7x^2 - 1x[/tex]

[tex]-(-7x^2 + 16x)[/tex]

[tex]-15x - 1[/tex]

[tex]-(-15x + 35)[/tex]

[tex]-36[/tex]

Therefore, the dividend is [tex]3x^3 + 0x^2 - 1x - 1[/tex], which is option A.

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Answer: 62,000

         2,000+4,000+8,000+16,000+32,000

Answer:

88 days

Step-by-step explanation:

inversely proportional means for a response y and input x, we have the formula

y = k/x

10 days = k/44 workers

multiply both sides by 44 to find k

44 workers * 10 days = k

k = 440 workers * days

y = k/x

x = 5 workers

440 workers * days / 5 workers = y

88 days = y

The interval over which the functions have the same average rate of change is given as follows:

x = 1 to x = 2.

The average rate of change of a function is given by the change in the output divided by the change in the input.

The function g(x) is given as follows:

g(x) = 18x + 12.

It is a linear function with a slope of 18, hence the average rate of change over each interval is of 18.

For function f(x), the rates are given as follows:

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y = 4. is the equation of the line in slope intercept form.

The formula for calculating the equation of a line in point slope form is expressed as:

y - y0 = m(x - x0)

where:

m is the slope

(x0, y0) is the point on the line

Using the coordinate points (-3, 4) and (2, 4). The slope is calculated as:

Slope = 4-4/2-(-3)
Slope = 0/5
Slope = 0

The required equation will therefore be expressed as:

y - 4 = 0(x - 2)

y - 4 = 0

y = 4

Hence the slope-intercept form of the equation is y = 4.

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Answer:

x = 7.5

m∠EGB = 75°

m∠EGA = 105°

Step-by-step explanation:

According to the Alternate Exterior Angles Theorem, angles EGB and CHF are equivalent. This means we can find x by setting them equal to each other.

10x = 2x + 60 (subtract 2x from both sides)

8x = 60 (divide both sides by 8)

x = 7.5

We can now plug in x to find m∠EGB.

2(7.5) + 60 = 15 + 60 = 75

Therefore, m∠EGB = 75°

As EGB and EGA are supplementary angles, that means that together, they add up to 180°. That means:

75 + EGA = 180 (subtract 75 from both sides)

m∠EGA = 105°