Consider a disease whose presence can be identified by carrying out a blood test. Let p denote the probability that a randomly selected individual has the disease. Suppose n individuals are independently selected for testing. One way to proceed is to carry out a separate test on each of the n blood samples. A potentially more economical approach, group testing, was introduced during World War II to identify syphilitic men among army inductees. First, take a part of each blood sample, combine these specimens, and carry out a single test. If no one has the disease, the result will be negative, and only the one test is required. If at least one individual is diseased, the test on the combined sample will yield a positive result, in which case the n individual tests are then carried out. [The article "Random Multiple-Access Communication and Group Testing"† applied these ideas to a communication system in which the dichotomy was active/idle user rather than diseased/nondiseased.] If p = 0.15 and n = 5, what is the expected number of tests using this procedure? (Round your answer to three decimal places.)

Final answer:

Normal distribution characterizes variables like the number of ants per acre, and the Central Limit Theorem helps understand the distribution of sample means. Allele frequencies are calculated by multiplying the number of homozygote ants by two and then dividing by the total number of alleles.

Explanation:

When discussing the number of ants per acre in a forest, you're dealing with a normal distribution, which is a probability distribution that is symmetric about the mean. With a known mean (μ) of 45,289 and a standard deviation (σ) of 12,340, if we let X represent the number of ants in a randomly selected acre, we can make various probabilistic predictions.

The Central Limit Theorem (CLT) applies when considering the sampling distribution of the sample mean. For example, if we have a population with a mean (μ) of 50 and standard deviation (σ) of 4, and take 100 samples each of size 40, the CLT tells us that the sampling distribution of the sample mean will be approximately normally distributed, centered around the population mean (μ), and with a standard deviation equal to the population standard deviation (σ) divided by the square root of the sample size (n). This is because, as per the law of large numbers, the sample means tend to get closer to the population mean as the sample size increases.

When calculating allele frequencies, it's essential to multiply the number of homozygote ants by 2 (as each ant has two alleles) to obtain the number of that allele in the population. The allele frequency is then the number of a specific allele divided by the total number of alleles.

Answer:

  y = 2x^2 -x -3

Step-by-step explanation:

It can be convenient to use the quadratic regression capability of a graphing calculator or spreadsheet. That's what we did in the attachment.

__

Fill in the given points in the equation to find three linear equations in a, b, c.

  3 = a(2^2) +b(2) +c

  7 = a(-2)^2 +b(-2) +c

  -2 = a(1^2) +b(1) +c

Subtracting the last equation from the first two gives ...

  (3) -(-2) = (4a +2b +c) -(a +b +c)   ⇒   5 = 3a +b

  (7) -(-2) = (4a -2b +c) -(a +b +c)   ⇒   9 = 3a -3b . . . . . [eq5]

Subtracting the second of these equations from the first gives ...

  (5) -(9) = (3a +b) -(3a -3b)   ⇒   -4 = 4b

  b = -1

Dividing [eq5] by 3 gives ...

  3 = a - b

  3 = a -(-1)

  2 = a

Using the original 3rd equation, we have ...

  -2 = a +b +c

  -2 = 2 +(-1) + c

  -3 = c . . . . . . . . subtract 1

The desired quadratic is ...

  y = 2x^2 -x -3

Answer:

  (a)  412 ft

  (b)  276 ft

Step-by-step explanation:

Consider the attached diagram.

(a) The internal angle of triangle RBT at B is 90° -10° = 80°. Since we know lengths RB and BT, we can find the length RT using the law of cosines:

  RT² = RB² +BT² -2·RB·BT·cos(80°) = 190² +400² -2·190·400·cos(80°)

  RT² ≈ 169,705.477

  RT ≈ √169,705.477 ≈ 411.95

The guy wire to the hillside should be about 412 feet long.

__

(b) The Pythagorean theorem can be used to find the shorter wire length.

  LM² = LB² +MB²

  LM = √(190² +200²) = √76,100

  LM ≈ 275.86

The guy wire to the flat side should be about 276 feet long.

Answer:

[tex]t=\frac{423-433}{\frac{2.6}{\sqrt{15}}}=-14.896[/tex]    

[tex]df=n-1=15-1=14[/tex]  

We need to find in the t distribution with df=14 a value who accumulates 0.1 of the area in the left and we got [tex]t_{crit}= -1.345[/tex].

Since our calculated value for the statistic is is so much lower than the critical value we have enough evidence to reject the null hypothesis, and we can conclude that the true mean for this case is significantly less than 433 and then the machine is underfilling.

Step-by-step explanation:

Data given

[tex]\bar X=423[/tex] represent the sample mean

[tex]s=26[/tex] represent the sample standard deviation

[tex]n=15[/tex] sample size  

[tex]\mu_o =433[/tex] represent the value that we want to test

[tex]\alpha=0.1[/tex] represent the significance level for the hypothesis test.

t would represent the statistic (variable of interest)  

[tex]p_v[/tex] represent the p value for the test

System of hypothesis

We need to conduct a hypothesis in order to check if the true mean is less than 433 (underfilling), the system of hypothesis would be:  

Null hypothesis:[tex]\mu \geq 433[/tex]  

Alternative hypothesis:[tex]\mu < 433[/tex]  

The statistic is:

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex]  (1)  

Calculate the statistic

[tex]t=\frac{423-433}{\frac{2.6}{\sqrt{15}}}=-14.896[/tex]    

Decision rule

The degrees of freedom are:

[tex]df=n-1=15-1=14[/tex]  

We need to find in the t distribution with df=14 a value who accumulates 0.1 of the area in the left and we got [tex]t_{crit}= -1.345[/tex]

Since our calculated value for the statistic is is so much lower than the critical value we have enough evidence to reject the null hypothesis, and we can conclude that the true mean for this case is significantly less than 433 and then the machine is underfilling.

Answer:

44

Step-by-step explanation:

Input 8 into x

f(8)=6(8)-4

f(8)=48-4

f(8)=44

Answer: [tex].47[/tex]

3 tenths is written as

Decimal: [tex].3[/tex]

Fraction: [tex]\frac{3}{10}[/tex]

17 hundredths is written as

Decimal: [tex].17[/tex]

Fraction: [tex]\frac{17}{100}[/tex]

Add

[tex]3/10+17/100[/tex]

[tex]=.47[/tex]

Fraction Form

[tex].47=\frac{47}{100}[/tex]

Answer:

The appropriate test statistic to test the claim that the mean for all car owners is less than 7.5 years is -1.997.

Step-by-step explanation:

We are given that 200 randomly selected car owners are surveyed, it is found that the mean length of time they plan to keep their car is 7.01 years, and the standard deviation is 3.47 years.

Let [tex]\mu[/tex] = mean for all car owners.

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \geq[/tex] 7.5 years     {means that the mean for all car owners is more than or equal to 7.5 years}

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] < 7.5 years    {means that the mean for all car owners is less than 7.5 years}

The test statistics that will be used here is One-sample t test statistics as we don't know about population standard deviation;

                                T.S.  = [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex]  ~ [tex]t_n_-_1[/tex]

where, [tex]\bar X[/tex] = sample mean length of time = 7.01 years

            s = sample standard deviation = 3.47 years

            n = sample of cars = 200

So, test statistics  =  [tex]\frac{7.01-7.5}{\frac{3.47}{\sqrt{200} } }[/tex]  ~ [tex]t_1_9_9[/tex]

                               =  -1.997

Hence, the appropriate test statistic to test the claim is -1.997.

Answer:

cluster sample

Step-by-step explanation:

Answer:

The sidewalk moves at 0.5 ft/sec

Josie's speed walking on a non-moving ground is 3.5ft/sec

Step-by-step explanation:

Let x represent the speed of the side walk and y represent her walking speed

It takes Jason's 8-year-old daughter Josie 44 sec to travel 176 ft walking with the sidewalk

Distance = speed × time

176 = (x+y)×44

44x+44y = 176

x+y = 4 .......1

It takes her 7 sec to walk 21 ft against the moving sidewalk in the opposite direction).

21 = (y-x)7

7y - 7x = 21

y - x = 3 ......2

Add equation 1 to 2

2y = 7

y = 3.5 ft/sec

From equation 1

x + y = 4

x = 4 - 3.5 = 0.5

x = 0.5 ft/sec

The sidewalk moves at 0.5 ft/sec

Josie's speed walking on a non-moving ground is 3.5ft/sec

Answer: Josie's speed walking on a non-moving ground is 3.5 ft/sec

The side walk moves at 0.5 Ft/sec

Step-by-step explanation:

Let x represent Josie's speed walking on a non-moving ground.

Let y represent the speed of the sidewalk.

It takes Jason's 8-year-old daughter Josie 44 sec to travel 176 ft walking with the sidewalk. It means that the total speed at which she moved is

(x + y) ft/sec

Distance = speed × time

Therefore,

176 = 44(x + y)

Dividing both sides by 44, it becomes

4 = x + y- - - - - - - - - - - - - -1

It takes her 7 sec to walk 21 ft against the moving sidewalk in the opposite direction). It means that the total speed at which she moved is (x - y) ft/sec

Therefore,

21 = 7(x - y)

Dividing both sides by 7, it becomes

3 = x - y- - - - - - - - - - - - - -2

Adding equation 1 and 2, it becomes

7 = 2x

x = 7/2 = 3.5 ft/sec

Substituting x = 3.5 into equation 2, it becomes

3 = 3.5 - y

y = 3.5 - 3 = 0.5 ft/sec

Answer:

33.66 cm

Step-by-step explanation:

Multiply to find the product.

In the plot, I have graphed two functions on the same set of axes.

The first function, y(x) = e^(-0.5x) * sin(2x), is represented by a solid blue line with a 2-point line width.

The second function, y(x) = e^(-0.5x) * cos(2x), is shown with a dashed red line with a 3-point line width. Both functions are evaluated for 100 values of x ranging from 0 to 10.

The solid blue line represents the sine function, and the dashed red line represents the cosine function.

The legend, title, axis labels, and grid have been included to make the plot more informative and visually appealing.

In this plot, you can observe the oscillatory behavior of both functions as they decay exponentially with decreasing x.

The legend distinguishes between the two functions, and the grid helps in reading the values accurately.

The choice of line widths and colors enhances the visibility of the two functions, making it easier to compare their behavior over the specified range of x values.

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Answer:

0.6848

Step-by-step explanation:

Mean of \hat{p} = 0.453

Answer = 0.453

Standard deviation of \hat{p} :

= \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.453(1-0.453)}{100}} = 0.0498

Answer = 0.0498

P(0.0453 - 0.05 < p < 0.0453 + 0.05)

On standardising,

= P(\frac{0.0453-0.05-0.0453}{0.0498} <Z<\frac{0.0453+0.05-0.0453}{0.0498})

= P(-1.0044 < Z < 1.0044) = 0.6848

Answer = 0.6848

Using the Central Limit Theorem, it is found that the mean of the sampling distribution of the sample proportions is 0.453.

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]

In this problem:

By the Central Limit Theorem, the mean is [tex]\mu = p = 0.453[/tex].

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Answer:

4

Step-by-step explanation:

The sum of the first n terms of a geometric series is:

S = a₁ (1 − r^n) / (1 − r)

Given n = 6, S = 15624, and r = 5:

15624 = a₁ (1 − 5^6) / (1 − 5)

a₁ = 4

complete question:

If the height and base of the parallelogram shown are each decreased by 2 cm, what is the area of the new parallelogram?

A parallelogram with a base of 10 centimetres and a height of 8 centimetres.

Answer:

area = 48 cm²

Step-by-step explanation:

A parallelogram is quadrilateral with 4 sides formed by 2 pair of parallel lines. The area of a parallelogram is represented as follows :

area of parallelogram =  B × H

where

B = breadth

H = height

According to the question the height and the base each reduced by 2 cm.

The new base = 10 - 2 = 8 cm

The new height = 8 - 2 = 6 cm

area = B × H

area = 8 × 6

area = 48 cm²

The level of production (x) that will maximize the profit is approximately 36 units.

The profit is the revenue (R(x)) minus the cost (C(x)). So, we have:

P(x) = R(x) - C(x)

Given Functions:

Cost Function: C(x) = 45,000 + 100x + x³ dollars

Revenue Function: R(x) = 4000x dollars

Substitute the given functions into the profit function:

P(x) = R(x) - C(x)

P(x) = 4000x - (45,000 + 100x + x³)

P(x) = 4000x - 45,000 - 100x - x³

Simplify the profit function:

P(x) = -x³ + 3900x - 45,000

Find the critical points by differentiating the profit function and setting it to zero:

P'(x) = -3x² + 3900

Set P'(x) = 0 and solve for x:

-3x² + 3900 = 0

-3x² = -3900

x²= 1300

x = ±√1300

x = ±36.06

Evaluate the second derivative (P''(x)) to determine if these critical points are maxima or minima:

P''(x) = -6x

Substitute the critical points (x = ±36.06) into P''(x):

P''(x)= -6(36.06)

= -216.36 (negative value)

Since the second derivative is negative at x ≈ ±36.06, it confirms that x = 36.06 is a maximum point.

So, the level of production (x) that will maximize the profit is approximately 36 units.

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To determine the level of production that maximizes profit, we calculate the profit function, differentiate it to find critical points, and then identify the maximum. The critical points are found by setting the first derivative of the profit function to zero and solving for x.

To determine the level of production x that will maximize the profit for the manufacturer, we first need to calculate the profit function. Profit, P(x), is the difference between total revenue, R(x), and total cost, C(x).

The profit function is defined as P(x) = R(x) - C(x). Using the given cost function C(x) = 45,000 + 100x + x^3 dollars and the revenue function R(x) = 4000x dollars, we get:

P(x) = 4000x - (45,000 + 100x + x^3)

This simplifies to:

P(x) = -x^3 + 3900x - 45,000

To find the production level that maximizes profit, we need to determine the critical points by differentiating P(x) and setting the derivative equal to zero. The derivative of P(x) is:

P'(x) = -3x^2 + 3900.

Setting P'(x) = 0 gives:

-3x^2 + 3900 = 0

Solving for x yields two critical points, but only one will maximize profit. We can find this maxima by testing which value of x gives the higher P(x) or by using the second derivative test. After finding the correct value of x, we round it to the nearest whole number as the final answer.

Answer:

[tex] E(X) = \sum_{i=1}^n X_i P(X_i) [/tex]

And replacing we got:

[tex] E(X) = 1*0.05 +2* 0.3 +3* 0.4 +4*0.25 = 2.85[/tex]

So we are going to expect about 2,85 automobiles for this case.

Step-by-step explanation:

For this case we define the random variable X as "number of automobiles lined up at a Lakeside Olds dealer at opening time (7:30 a.m.)" and we know the distribution for X is given by:

X         1         2       3         4

P(X)  0.05  0.30  0.40   0.25

The expected value of a random variable X is the n-th moment about zero of a probability density function f(x) if X is continuous, or the weighted average for a discrete probability distribution, if X is discrete

For this case we can calculate the epected value with this formula:

[tex] E(X) = \sum_{i=1}^n X_i P(X_i) [/tex]

And replacing we got:

[tex] E(X) = 1*0.05 +2* 0.3 +3* 0.4 +4*0.25 = 2.85[/tex]

So we are going to expect about 2,85 automobiles for this case.

The expected number of automobiles at Lakeside Olds at opening time can be calculated as a weighted average, resulting in an expectation of approximately 3 cars.

The expected number of automobiles lined up at Lakeside Olds at opening time would be calculated by multiplying each number of automobiles by its respective probability and then adding up those products. This is essentially calculating a weighted average.

So for the given data:

Add up those results: 0.05 + 0.60 + 1.20 + 1.00 = 2.85. Thus, on a typical day, Lakeside Olds should expect approximately 3 automobiles to be lined up at opening time.

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Answer:

D. 144 cubic meters

Step-by-step explanation:

Volume of triangular prism(V)= Area of a triangle(A)×height of the prism(H)

That is,

V=AH

Where,

A=Area of a triangle

H=Height of the prism

Given,

A=1/2× base×height=16 square meters

Height=9 meters

Therefore,

Volume of triangular prism=16 square meters×9 meters

=144 cubic meters

Answer: D. 144 cubic meters

Step-by-step explanation:

A triangular prism consists of 2 triangular faces(base) and 3 rectangular faces.

The formula for determining the volume of a triangular prism is expressed as

Volume = area of the triangular base × height of the prism

Area of triangular base = 1/2 base × height

From the information given,

Height of prism = 9 meters

Area of triangular base = 16 square meters

Volume of the triangular prism = 16 × 9 = 144 cubic meters

Answer:

i believe that the answer is A.) but im not 100% sure, im about  65% sure

Step-by-step explanation:

The value of the expression x/z is 3/5.

Simplification of an expression is the process of writing an expression in the most efficient and compact form without affecting the value of the original expression.

For the given situation,

The expressions are

7x=3y ------ (1)

5y=7z ------ (2)

From equation 1,

[tex]7x=3y[/tex]

⇒ [tex]x=\frac{3y}{7}[/tex]

From equation 2,

[tex]5y=7z[/tex]

⇒ [tex]z=\frac{5y}{7}[/tex]

Now, [tex]\frac{x}{z}= \frac{\frac{3y}{7} }{\frac{5y}{7} }[/tex]

⇒ [tex]\frac{x}{z}=(\frac{3y}{7})(\frac{7}{5y} )[/tex]

⇒ [tex]\frac{x}{z}=(\frac{3}{5})[/tex]

Hence we can conclude that the value of the expression x/z is 3/5.

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The 95% confidence interval for the population mean based on a sample of 100 with mean 16 and population standard deviation 3 is (15.412, 16.588). For a margin of error not exceeding 0.5, a sample size of 139 is needed.

To compute the 95% confidence interval for the population mean μ, we use the formula for a confidence interval which is ȳ ± Z*(σ/√n), where ȳ is the sample mean, σ is the population standard deviation, n is the sample size, and Z is the z-score corresponding to the desired confidence level (for a 95% confidence interval, Z = 1.96).

Plugging the given values into the formula, we get 16 ± 1.96*(3/√100), which simplifies to 16 ± 0.588. Thus, the 95% confidence interval for the population mean μ is (15.412, 16.588).

For the second part of the question, the formula used to find the sample size needed for a certain margin of error (E) at a certain confidence level is n = (Zσ/E)^2. Substituting the given values into this formula, we get n = (1.96*3/0.5)^2 which is equal to 138.384. Since we can't have a fraction of a sample, we round this up to the nearest whole number, so we would need a sample size of 139 to estimate the population mean μ with a margin of error not exceeding 0.5 with a 95% confidence interval.

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To compute a 95% confidence interval for the population mean μ, we can use the formula: Confidence Interval = Sample Mean ± (Z * σ/√n). The 95% confidence interval for the population mean μ is (15.412, 16.588). To estimate the sample size needed to keep the margin of error within 0.5 with a 95% confidence level, we can use the formula: n = (Z^2 * σ^2) / (E^2). We need a sample size of at least 24 to estimate the population mean μ with a margin of error not exceeding 0.5, with a 95% confidence level.

To compute a 95% confidence interval for the population mean μ, we can use the formula:

Confidence Interval = Sample Mean ± (Z * σ/√n)

Where Z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.

For this problem:

Confidence Interval = 16 ± (1.96 * 3/√100)

Simplifying, we get:

Confidence Interval = 16 ± 0.588

Therefore, the 95% confidence interval for the population mean μ is (15.412, 16.588).

To estimate the sample size needed to keep the margin of error within 0.5 with a 95% confidence level, we can use the formula:

n = (Z^2 * σ^2) / (E^2)

Where Z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and E is the maximum acceptable margin of error.

For this problem:

n = (1.96^2 * 3^2) / (0.5^2)

Simplifying, we get:

n = 23.532

Therefore, we need a sample size of at least 24 to estimate the population mean μ with a margin of error not exceeding 0.5, with a 95% confidence level.

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Answer:

21 weeks.

Step-by-step explanation:

To start, you have to get the pounds lost per week.

12/3=4

4 pounds lost per week.

Next, find out how many weeks it takes by solving this equation,

4x=84

Divide 84/4=21

Therefore, it took 21 weeks to lose 84 pounds.

Step-by-step explanation:

"Of the following dotplots, which best represents the possible results from the simulation described?"

The sample size is 100, and the probability of success is 0.015, so the expected value is 1.5.  Meaning we would expect a dotplot with most of the dots at 1 and 2.

By considering a simulation with 50 trials designed to estimate the number of canceled flights from a random sample of size 100, where the probability of success, a canceled flight, is 0.015. A dot-plot with most of the dots at 1 and 2.

Given:

Sample size (n) = 100

Probability = 0.015

To estimate the number of canceled flights

Expected value = 0.015 x 100

Expected value = 1.5

Therefore, a dot-plot with most of the dots at 1 and 2.

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Answer:

140 degrees

Step-by-step explanation:

Since this triangle is isosceles, the base angles must be equal. This means that two out of the three angles in the triangle are 20 degrees. Since all of the angles together in a triangle must add up to 180 degrees, the angle of the vertex is 180-20-20=140 degrees. Hope this helps!

Answer:

140

Step-by-step explanation:

The base angle is 20.  That means the other base angle is also 20

20+20 = 40

The sum of the angles of a triangle is 180

180-40= 140

That means the third angle must be 140

Answer:

4/5

Step-by-step explanation:

.8 is a fourth of 1.0

.20

.40

.60

.80

1.0

since there are five total numbers from that sequence, and .8 is the fourth, the simplest form it could go to is 4/5

The decimal number 0.8 as a fraction in the simplest form is 4/5.

In Mathematics and Geometry, a fraction simply refers to a numerical quantity (numeral) which is not expressed as a whole number. This ultimately implies that, a fraction is simply a part of a whole number.

In this exercise and scenario, we would convert the given decimal number into a fraction as follows;

0.8 = 8/10

By dividing both the numerator and denominator by 2, we have the following

8/10 = 4/5

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Answer:

FALSE. If he is risk-loving, he will rather bet $1,000 rather than $10.

Step-by-step explanation:

FALSE.

As Paul is risk-loving, he will take more risk, even if there is no more chances of winning if he bets $10 or $1,000. He will focus on the probability of winnings rather than the expected losses.

In this case, the probabilities are the same independently of the amount that Paul bets.

Answer:

x/2=5+4=9

x=9*2=18

x=18!

Answer:

1. Change each number to an improper fraction.

2. Simplify if possible.

3. Multiply the numerators and then the denominators.

4. Put answer in lowest terms.

Step-by-step explanation:

Answer:

[tex]x=-1[/tex]

Step-by-step explanation:

Equations

The following equation will be solved

[tex]\displaystyle \frac{2}{x+3}-\frac{3}{4-x}=\frac{2x-2}{x^2-x-12}[/tex]

Changing signs of the second term on the left side

[tex]\displaystyle \frac{2}{x+3}+\frac{3}{x-4}=\frac{2x-2}{x^2-x-12}[/tex]

Operating

[tex]\displaystyle \frac{2(x-4)+3(x+3)}{x^2-x-12}=\frac{2x-2}{x^2-x-12}[/tex]

Simplifying denominators, provided

[tex]x^2-x-12\neq 0[/tex]

[tex]2(x-4)+3(x+3)=2x-2[/tex]

Operating

[tex]2x-8+3x+9=2x-2[/tex]

Solving

[tex]\boxed{x=-1}[/tex]

Since

[tex](-1)^2-(-1)-12=-10\neq 0[/tex]

Solution (1)