find all $r$ for which the infinite geometric series \[2 6r 18r^2 54r^3 \dotsb\]is defined. enter all possible values of $r,$ as an interval.

The triangle has an area of 75 units² then the area of triangle be 15 units.

The space bounded by a triangle's three sides is known as the area of a triangle. The total area that is bounded by a triangle's three sides is referred to as the triangle's area.

Area of triangle = 1/2 (b h), where b is the base and h is the height, is the fundamental formula for calculating a triangle's area. The type of the triangle and the known dimensions can affect other formulas that are used to calculate the area of a triangle.

Let x be base of triangle

height = 19 units

Area of triangle = 1/2 base × height

substitute the values in the above equation, we get

75 = 1/2 × x × 10

simplifying the equation, we get

75 = x × 5

x = 75/5

x = 15 units

Therefore, the area of triangle be 15 units.

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Based on the information, it can be inferred that Mhlaba food processor on offer is the right choice for Mhlaba due to its quality product.

According to the information, Mhlaba should buy a home food processor because he only needs to make small portions for himself and his family.

Therefore, there is no need for Mhlaba to look for another more professional food processor since he will not work as a cook nor will he require a large-capacity machine. Then he should choose the one on offer.

Note: This question is incomplete. Here is the complete information:

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The measure of <OXY is 90 degree.

Tangents to circles are lines that cross the circle at a single point. Point of tangency refers to the location where a tangent and a circle converge. The circle's radius, where the tangent intersects it, is perpendicular to the tangent.

Given:

As, A line that crosses a circle at precisely one point, known as the point of tangency or tangency point, is said to be tangent to the circle.

The fact that the radius is perpendicular to the tangent line from the circle's center to the point of tangency is a significant outcome.

So, according to this tangent XY toucehes the circle at point X.

Thus, <OXY is perpendicular i.e., <OXY = 90

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A coterminal angle of θ such that 0 ≤ θ ≤ 2π is 3π/4

A. A coterminal angle of θ such that 0 ≤ theta  ≤ 2π is

= theta + 2π

= -11π/4 + 2π

= 3π/4.

B. The exact values of the six trigonometric functions evaluated at θ are as follows:

sin(θ) = sin(-11π/4) = -√2/2

cosec (θ) = cosec (-11π/4) = -2/√2

cos(θ) = cos(-11π/4) = -√2/2

sec (θ) = sec (-11π/4) = -√2/2

tan(θ) = tan(-11π) = 1

cotan (θ) = cotan (-11π) = 1

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The smallest number of blocks needed to build the wall is 350 blocks which are 1 foot high and either 2 feet long.

A rectangle is a two-dimensional shape where the length and width are different.

The area of a rectangle is given as:

Area = Length x width

We have,

Dimension of the wall.

Length = 100 feet

Height = 7 feet

Area = 100 x 7 = 700 ft²

Dimension of the larger block.

Length = 2 feet

Height = 1 feet

Area = 2 x 1 = 2 ft²

Dimension of the smaller block.

Length = 1 feet

Height = 1 feet

Area = 1 x 1 = 1 ft²

Now,

The number of smaller blocks needed to build the wall.

= 700

The number of larger blocks needed to build the wall.

= 700 / 2

= 350

Thus,

The smallest number of blocks needed is 350 blocks which are 1 foot high and either 2 feet long.

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Answer:

Therefore, you will need to order approximately 9.07 cubic yards of topsoil to fill the garden to a depth of 6 inches.

Step-by-step explanation:

To calculate the amount of topsoil needed, we need to find the volume of the garden in cubic yards, since topsoil is sold by the cubic yard.

The garden is 35 feet by 14 feet and 6 inches (or 0.5 feet) deep.

Volume of the garden = length x width x depth

= 35 feet x 14 feet x 0.5 feet

= 245 cubic feet

To convert cubic feet to cubic yards, we divide by 27 (since 1 cubic yard = 27 cubic feet):

Volume in cubic yards = 245 cubic feet / 27

= 9.07 cubic yards (rounded to two decimal places)

Therefore, you will need to order approximately 9.07 cubic yards of topsoil to fill the garden to a depth of 6 inches.

Answer:

3.2 × 10⁻⁸

Step-by-step explanation:

Scientific notation is written in the form [tex]a \times 10^n[/tex], where [tex]1 \leq a < 10[/tex] and [tex]n[/tex] is any positive or negative whole number.

To convert a number into scientific notation, move the decimal point to the left or right until there is one non-zero digit to the left of the decimal point.

The number of times you have moved the decimal point is the power of 10 (the value of n).

Therefore, to convert 0.000000032 to scientific notation, move the decimal point 8 places to the right so that [tex]a[/tex] is 3.2.

As the decimal point has been moved 8 places the right, the exponent of 10 is -8.

Therefore,  0.000000032 = 3.2 × 10⁻⁸.

Answer:

Step-by-step explanation:

Let's say the enrollment in the Spanish class is x.

According to the information given, the enrollment in the French class is 0.72 of the enrollment in the Spanish class, which means the enrollment in the French class is 0.72x.

The enrollment compared to Spanish class as a fraction is 0.72x / x = 0.72.

The enrollment compared to Spanish class as a percent is 0.72 x 100% = 72%.

The solution is, The enrollment compared to Spanish class as a percent is

72%.

Here, we have,

Let's say the enrollment in the Spanish class is x.

According to the information given, the enrollment in the French class is 0.72 of the enrollment in the Spanish class, which means the enrollment in the French class is 0.72x.

The enrollment compared to Spanish class as a fraction is,

0.72x / x = 0.72.

The enrollment compared to Spanish class as a percent is,

0.72 x 100% = 72%.

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A: The expression to represent the average change in temperature is -11/10 °F per minute.

B: The average change in temperature is -1.1 °F per minute,

C:  For the second attempt average change is -0.1 per minute.

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. An expression's structure is as follows:

Expression: (Math Operator, Number/Variable, Math Operator)

Given temp. at beginning = 23°F

He monitors the temperature of the solution every minute for 10 minutes, and

He records the change in temperature from his previous reading. His data show a decrease of 2 degrees Fahrenheit four times, a decrease of 3 degrees Fahrenheit three times, no temperature change two times, and an increase of 6 degrees Fahrenheit

A) The expression to represent the average change in temperature each minute is given as follows:

The observations are given as follows:

-2, -2, -2, -2, -3, -3, -3, 0, 0, 6.

Mean = (4 x (-2) + 3 x (-3) + 0(2) + 6)/10.

Mean = -11/10

B) The average change in temperature for the first trial is given as follows:

The observations are given as follows:

-2, -2, -2, -2, -3, -3, -3, 0, 0, 6.

The average change  = -11/10

The average change  =  -1.1ºF per minute.

C) In the second attempt of the experiment, the average change in temperature per minute,

A data set's mean is computed as the total number of observations divided by the total number of observations in the data set.

There are 10 observations in total in this problem.

The observations are given as follows:

-2, -2, -2, -2, -3, -3, -3, 0, 0, 6.

Hence, the mean is calculated as follows:

Mean = (4 x (-2) + 3 x (-3) + 6)/10

Mean = -1.1ºF per minute

1ºF greater than this for the second trial is given as follows:

-1.1 + 1 = -0.1 ºF per minute.

Hence the average change in temperature is -1.1 °F per minute,

C: For the second attempt average change is -0.1 per minute.

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The question is incomplete. The complete question is 'Edgar measures the temperature of a solution at the beginning of an experiment to be 23 degrees Fahrenheit . He monitors the temperature of the solution every minute for 10 minutes. He records the change in temperature from his previous reading. His data show a decrease of 2 degrees Fahrenheit four times, a decrease of 3 degrees Fahrenheit three times, no temperature change two times, and an increase of 6 degrees Fahrenheit once. (a) Write an expression to represent the average change in temperature each minute. (b) Using your expression from Part (a), determine the average change in temperature for the first trial(c) Edgar repeats the experiment and finds the average change in temperature each minute is 1 degrees Fahrenheit greater in the second trial than in the first trial. What is the average change in temperature each minute in the second trial of the experiment?'

Answer:

Step-by-step explanation:

PART A: No, the trinomial 4x^3 + 28x + 49 is not a perfect square trinomial. To be a perfect square trinomial, the first and last terms must be perfect squares and the middle term must be twice the product of the square roots of the first and last terms. In this case, the first and last terms are not perfect squares, and the middle term is not twice the product of the square roots of the first and last terms.

PART B: To factor the expression 6x^2 + 5x - 4, we need to find two binomials whose product is equal to this expression.

First, we can break down the coefficient of x into two numbers whose product is 6 * (-4) = -24 and whose sum is 5. These numbers are 8 and -3.

Next, we use these two numbers to split the middle term 5x into two terms:

6x^2 + 8x - 3x - 4

We can now factor the expression by grouping:

(6x^2 + 8x) - (3x + 4)

2x(3x + 4) - 1(3x + 4)

(2x - 1)(3x + 4)

Therefore, the area of the rectangle can be factored as (2x - 1)(3x + 4).

To determine the dimensions of the rectangle, we set each factor equal to one of the dimensions and solve for the other dimension:

2x - 1 = length

3x + 4 = width

Alternatively, we could switch the factors and get the same answer:

2x - 1 = width

3x + 4 = length

Either way, these are the dimensions of the rectangle, expressed in terms of x.

Rows of alpha are numbered 0...24 and columns are numbered 0...9 is the true statement about alpha.

What is alpha?

The first letter of the Greek alphabet is known as alpha. Alpha is typically used to signify an unidentified angle. Gamma, alpha, and beta are utilized as variables just as x, y, and z in mathematics. Mainly in quadratic problems, it is employed.

Since symbols are only meant for communication, you can create whatever you choose with them. It may take the form of a real number, vector, tensor, or index.

The rows of the alpha have to start from 0 so, the true statement lies in option a and option b.

Also, the columns are to be numbered till 9. So, the true statement lies in option a.

Hence, the true statement is that rows of alpha are numbered 0...24 and columns are numbered 0...9.

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The proportion of oil tankers that had spills is:

1 - (445/563) = 0.208

Rounded to three decimal places, the answer is 0.208. Alternatively, this can be expressed as the fraction 52/253.

On solving the provided question, we can say that by bodmas, Hardwood cost = 5.59*320 =  $1788.8 ; laminate cost = 2.89*320 = $924.8 and difference = 1788.8-924.8 = $864

The abbreviation BODMAS aids kids in recalling the hierarchy of mathematical operations (the correct order for solving math problems). The acronym BODMAS is made up of the following terms: B Bracket, O Degree (Power/Exponent or Root), D Division, M Multiplication, A Addition, and S Subtraction. BODMAS is also referred to as PEDMAS in some locales. This symbolizes exponents, division, multiplication, addition, and subtraction. It also represents parentheses. The BODMAS rule states that parentheses must be handled first, followed by powers or roots (i.e., of), divisions, multiplications, additions, and lastly subtractions. While he is known as BODMAS in India and the UK, PEMDAS is primarily utilized in the US. However, they are identical to one another. The parentheses, order, and order of the operations of addition, subtraction, multiplication, and division are the same for the two rules.

A room addition requires 320 sq ft

Hardwood costs $5.59 per sq ft.

laminate costs $2.89 per sq ft.

Hardwood cost = 5.59*320 =  $1788.8

laminate cost = 2.89*320 = $924.8

difference = 1788.8-924.8 = $864

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The coefficients of the product are 25 and -30. Option A nd E

These are expressions that are composed of arithmetic or mathematical operations, such as;

They are also made up of variables, coefficients, terms, constants and factors.

Coefficients are numbers in the front of variables in the expression.

Given that;

(5y − 3)(5y -3 )

expand the bracket

25y² - 15y - 15y + 9

25y²  -30y + 9

Hence, the coefficients are 25 and - 30

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Answer:

Year | Rate | Interest (9%) | Principal Invested | Final Value

1 | 9% | 52.20 | 580.00 | 632.20

2 | 9% | 56.94 | 580.00 | 688.14

Answer:

0.5

Step-by-step explanation:

6 mins & travelled 3 miles farther

so miles/min = 3/6 = 0.5 mile(s) per minute.

None of them is correct and the value of f(5/2) is 1.6 to 1 decimal place

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

f(x) = √x

Substitute 5/2 for x

So, we have

f(5/2) = √5/2

When the above is compared to their functions, we have no equivalent expression

This means that none of the notations is correct

Next. we have

f(5/2) = 1.6

Hence, the value is 1.6

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For the function, [tex]f(x) = \frac{1 - x^2}{x - 1}[/tex] the Limit of f(x) as x approaches 1 does not exist.

A function can be defined as the outputs for a given set of inputs.

The inputs of a function are known as the independent variable and the outputs of a function are known as the dependent variable.

Given, [tex]f(x) = \frac{1 - x^2}{x - 1}[/tex], [tex]g(x) = \frac{x^2 - 5x + 6}{x - 3}[/tex].

We know, In a rational expression (p/q), q ≠ 0 as the expression would be undefined.

Therefore, The function for which the Limit of f(x) as x approaches 1 does not exist is [tex]f(x) = \frac{1 - x^2}{x - 1}[/tex], as at x = 1, f(x) is undefined.

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The correct statement regarding the exponential functions is given as follows:

Since y-intercepts indicate the amount of each bacteria at the start of the experiment, there was more of bacteria B than bacteria A at the start.

An exponential function is defined as follows:

y = ab^(x/n).

In which the parameters are defined as follows:

For Bacteria A, we have that:

For Bacteria B, we have that:

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1. The algae blooms was larger initially is the first algae.

2. The first algae bloom is decreasing more quickly than the second algae bloom.

1. To determine which algae bloom was larger initially, we can compare the values of the functions at d=0.

For the first algae bloom, we have A(0) = 200.

For the second algae bloom, we have B(0) = 100*27⁰ = 100.

Therefore, the first algae bloom was larger initially.

2. To determine which algae bloom is decreasing more quickly, we can compare the rates of decrease of the two functions. The rate of decrease is given by the derivative of the function, which tells us how fast the area is changing with respect to time.

For the first algae bloom, we have A'(t) = -20e^(-0.2t).

For the second algae bloom, we have B'(d) = -100*27^(-1)d.

To compare these rates of decrease, we can evaluate them at the same point in time or days. Let's evaluate them at d=1, which means one day has passed since the first measurement.

For the first algae bloom, we have A'(1) = -20e^(-0.2) ≈ -15.24.

For the second algae bloom, we have B'(1) = -100*27^(-1) ≈ -3.70.

Therefore, we can see that the first algae bloom is decreasing more quickly than the second algae bloom. This is because the rate of decrease of the first bloom is more negative than the rate of decrease of the second bloom.

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Let's start by setting up an equation to represent the situation:

59 + 57x = 15 + 79x

where x is the number of months it will take for both Thomas and Christian to have the same amount of money in their accounts.

Simplifying this equation, we get:

64x = 44

Dividing both sides by 64, we get:

x = 0.6875

So it will take approximately 0.6875 months for Thomas and Christian to have the same amount of money in their accounts. However, since time is measured in months, we cannot have a fraction of a month. Therefore, we round up the result to the nearest whole month and conclude that it will take 1 month for both Thomas and Christian to have the same amount of money in their accounts.

You can deduce from your facts that assertion A is correct. Statement B is not conclusive. Statement C is incorrect. Statement D is not conclusive. statement E is false.

The field of mathematics concerned with probability is known as probability theory. Although there are various distinct interpretations of probability, probability theory approaches the idea rigorously mathematically by articulating it through a set of axioms.

Here,

From your data, you can conclude that:

A. A parent-offspring regression for trait C will have the highest slope.

Heritability is the proportion of variation in a trait that can be explained by genetic variation. A higher heritability indicates that more of the variation in a trait is due to genetic differences, and less is due to environmental differences. If the heritability of a trait is 0.8, this means that 80% of the variation in the trait is due to genetic differences, and a parent-offspring regression would have a steeper slope than if the heritability was lower.

B. Selection of trait A will have no effect on the population mean for this trait.

This statement cannot be concluded from the data. Heritability of a trait does not dictate the responsiveness of a trait to selection. Even if the heritability of a trait is low, it can still respond to selection, and even if the heritability of a trait is high, it may not respond much to selection, depending on the presence of other factors such as the distribution of the trait in the population, the presence of linkage disequilibrium, and the presence of genetic constraints.

C. A parent-offspring regression for trait B will be a flat line.

This statement is false. If the heritability of a trait is 0.5, this means that half of the variation in the trait is due to genetic differences, and half is due to environmental differences. A parent-offspring regression would not be a flat line, but would still have a slope, albeit not as steep as if the heritability was higher.

D. Selection on each behavior will result in similar changes in the population mean for each trait.

This statement cannot be concluded from the data. Heritability is only one of many factors that affects the responsiveness of a trait to selection, and the responsiveness of each trait will depend on other factors as well.

E. A parent-offspring regression on trait A will have a slope of 0.9.

This statement is false. Heritability is the proportion of variation in a trait that can be explained by genetic differences, not the slope of a regression. The slope of a parent-offspring regression would depend on other factors such as the distribution of the trait in the population and the presence of environmental factors.

From your data, you can conclude that: statement A is true. statement B cannot be concluded. statement C is false. statement D cannot be concluded. statement E is false.

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Correct answer to the question is: operating profit will be the same on both statements; marketing and administrative costs will be the same on both statements.

A gross margin income statement focuses on the difference between the cost of goods sold (COGS) and the selling price of the products. It shows the gross profit as a percentage of sales, which can be used to evaluate the company's ability to generate revenue and cover its costs.

Both the gross margin and the contribution margin income statements provide important information to the company, but they offer different perspectives on the company's profitability. The gross margin income statement is useful for evaluating the overall efficiency of the company's sales and production processes, while the contribution margin income statement is useful for evaluating the potential profit from selling additional units.

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The complete question is :

Assume total units produced equals total units sold. If the company prepares a gross margin income statement and a contribution margin income statement, _____.

Answer: x = -22/7

Step-by-step explanation: Basically, all angles are the same as indicated in the image, this means that the triangle is equilateral and that all sidelengths will  be the same lengths

we can now set up an equation that equals both side lengths to find x

4x+16 = -3x -6

4x+16-16= -3x -6 -16

4x = -3x - 22

4x +3x= -3x + 3x -22

7x/7 = -22/7

x = -22/7

For the function y=f(x), the domain is (-2,2] and the range is (-2,3).

The corresponding values are -

f(-1) = -6, f(0) = -5, f(1) = 6, f(2) = 3.

What is a function?

In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.

The graph for function y = f(x) is given.

The collection of input or argument values for which a function is actual and defined is known as its domain.

In the graph it can be seen that the graph is intersecting x-axis at x = -2 and at x = 2.

So, the domain is (-2,2]. The ( symbol denotes empty circle whereas the ] symbol denotes a filled circle.

A function's range is the collection of values for which the dependent variable is defined.

The coordinates for the ending of the graph is (2,3). So, the range of the function is [-2,3].

The value for f(-1) as seen in the graph is -

x = -1, then y = -6.

The value for f(0) as seen in the graph is -

x = 0, then y = -5.

The value for f(0) as seen in the graph is -

x = 1, then y = 6.

The value for f(2) as seen in the graph is -

x = 2, then y = 3.

Therefore, the domain and range is (-2,2] and (-2,3) respectively.

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f⁻¹(x) = 4x + 7

This can be solved using the concept of function.

Function is a mathematical phrase, rule, or law that establishes the connection between an independent variable and a dependent variable (the dependent variable). A function is described as a relationship between a group of inputs and one output each. In simple terms, a function is a connection between inputs, with each input corresponding to exactly one output. Every function has its own domain and codomain, as well as a range. f(x), where x is the input, is a common way to refer to a function.

Given that,

f(x) = x -7/4

f⁻¹(x) = 4x + 7

Now, f (f⁻¹(x)) = f (4x + 7)

=  [{(x - 7) / 4} ÷ 4] + 7

= x          [Proved]

Next, f⁻¹(f(x))

= f⁻¹ (x -7/4)

= 4 × [(x -7/4)] + 7

= x          [Proved]

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finance charge Jack will​ pay is $1,512.

A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.

Given, Jack Fitzgerald wishes to purchase a used car that has a cash price of ​$11,000. The installment terms include a down payment of ​$2,000 and 48 monthly payments of ​$219.

a)

The amount that needs to be financed is = 11,000 - 2,000 = $9,000.

Amount repayment = 219 x 48 = $8,100.

=> Finance charge = Amount repayment - Amount needs to be financed

Finance charge = 10512 - 9000

Finance charge = 1,512$

b)

We apply the PV for annuity formula to find the interest rate per one month denoted as i:

Amount needs to be finance = (equal monthly installment x i) / [ 1 - (1+i)^-36 ]

or 9,000 = (219/i) x [1 - (1+i)^-48 ] <=> i = 0.911%.

APR = 12 x i

APR = 10.932% .

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The solution to the equation x - 25 = 2 is x = 27.

What is the equation?

An equation is a mathematical statement that asserts the equality of two expressions. In other words, it is a statement that states that two values are equal.

To solve the equation x - 25 = 2, we first add 25 to each side of the equation:

x - 25 + 25 = 2 + 25

Simplifying the right side of the equation, we get:

x = 27

Hence, The solution to the equation x - 25 = 2 is x = 27.

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The multiplication inequality that shows the number of bags (b) the wheelbarrow can carry is 20b ≤ 300.

It shows a relationship between two numbers or two expressions.

There are commonly used four inequalities:

Less than = <

Greater than = >

Less than and equal = ≤

Greater than and equal = ≥

We have,

The wheelbarrow can carry upto = 300 lbs

The bag of soil weighs = 20 lbs.

Now,

Number of bags of soil = b

We can write an inequality for the number of bags of soil the wheelbarrow can carry.

20b ≤ 300

Thus,

20b ≤ 300 is the multiplication inequality to show the number of bags (b) the wheelbarrow can carry.

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The equation for Harper's situation is 10x = 150 + 5x. The number of t-shirts Harper needs to sell so that the amount of money she spends to start her business and the amount of money she earns are the same is 30 t-shirts.

What is an equation?

A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").

The price of one plane t-shirt is = $5

The money spent on printing equipment is = $150

The price for printed t-shirt is = $10

Let the number of t-shirts to be sold be x.

The expression for earning money is 10x

The amount she spends on starting the business is -

150 + 5x

So, according to question the equation is -

10x = 150 + 5x

10x - 5x = 150

5x = 150

x = 150 / 5

x = 30

Therefore, 30 t-shirts needs to be sold.

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