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Please help with #15
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The tangent and normal lines of the curve:

Case (i): y = (4 / 5) · x + 13 / 5

Case (ii): (x, y) = (0, 13 / 5)

Case (iii): y = - (5 / 4) · x + 35 / 4

In this problem we have the representation of a curve whose equations for tangent and normal lines must be found. Lines are expressions of the form:

y = m · x + b

Where:

Both tangent and normal lines are perpendicular, the relationship between the slopes of the two perpendicular lines is:

m · m' = - 1

Where:

The slope of the tangent line is found by evaluating the first derivative of the curve at intersection point.

Case (i) - First, determine the slope of the tangent line:

y = √(8 · x + 1)

y' = 4 / √(8 · x + 1)

y' = 4 / √25

y' = 4 / 5

Second, determine the intercept of the tangent line:

b = y - m · x

b = 5 - (4 / 5) · 3

b = 5 - 12 / 5

b = 13 / 5

Third, write the equation of the tangent line:

y = (4 / 5) · x + 13 / 5

Case (ii) - Find the coordinates of the intercept of the tangent line:

(x, y) = (0, 13 / 5)

Case (iii) - First, find the slope of the normal line:

m' = - 1 / (4 / 5)

m' = - 5 / 4

Second, determine the intercept of the normal line:

b = y - m' · x

b = 5 - (- 5 / 4) · 3

b = 5 + 15 / 4

b = 35 / 4

Third, write the equation of the normal line:

y = - (5 / 4) · x + 35 / 4

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Answer:

8

Step-by-step explanation:

Substitute the values in the expression, we have:

[tex]\displaystyle{|-1-3|+4}[/tex]

Evaluate:

[tex]\displaystyle{|-4|+4}[/tex]

Any real numbers in the absolute sign will always be evaluated as positive values. Thus:

[tex]\displaystyle{|-4|+4 = 4+4}\\\\\displaystyle{=8}[/tex]

Hence, the answer is 8. A quick note that z-value is not used due to lack of z-term in the expression.

The algebraic rule for translating the figure 3 units left and 2 units up is (x-3, y+2).  Option B.

To translate a figure 3 units to the left and 2 units up, we need to adjust the coordinates of the figure accordingly. The algebraic rule that represents this translation can be determined by examining the changes in the x and y coordinates.

When we move a figure to the left, we subtract a certain value from the x coordinates. In this case, we want to move the figure 3 units to the left, so we subtract 3 from the x coordinates.

Similarly, when we move a figure up, we add a certain value to the y coordinates. In this case, we want to move the figure 2 units up, so we add 2 to the y coordinates.

Taking these changes into account, we can conclude that the algebraic rule for translating the figure 3 units left and 2 units up is (x-3, y+2). The x coordinates are shifted by subtracting 3, and the y coordinates are shifted by adding 2. SO Option B is correct.

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Answer:

There is one solution. The solution is 2, 18, 19.

Step-by-step explanation:

If you want me to show working tell me in the comments and I'll edit the answer

Answer:

A. (2, 18, -19)

Step-by-step explanation:

To solve:

Z is the most suitable variable to remove first

Add the first equation to the second equation: (this conveniently removes both y and z)

(x+y-z) + (4x-y+z) = 1+9

Simplify

5x = 10

Solve

x = 2

Multiply the second equation by 2 and minus it to the third equation: (Solve for y)

2(4x-y+z) - (x-3y+2z) = 2(9) - (-14)

Simplify

8x-2y+2z-x+3y-2z=18+14

7x+y=32

Substitute using x=2

7(2) + y = 32

y = 32 - 14

y = 18

Now substitute x and y for their respective values into Equation 1

2 + (-18) - z = 1

Simplify

-z = 19

z = -19

So :

x = 2, y = 18 , z = -19

Answer:

To find the length of AB using the segment addition postulate , we need to add the lengths of segments AC and CB.

AC + CB = AB

Substituting the given lengths:

2x-3 + 24 = 5x+6

Simplifying and solving for x:

21 = 3x

x = 7

Now that we know x, we can substitute it back into the expression for AB:

AB = 2x-3 + 24 = 2(7)-3 + 24 = 14-3+24 = 35

Therefore, the length of AB is 35.

Step-by-step explanation:

The number line that represents the solution set for the inequality 3(8 – 4x) < 6(x – 5) is option C: A number line from negative 5 to 5 in increments of 1. An open circle is at negative 3, and a bold line starts at negative 3 and is pointing to the left.

To determine the solution set for the inequality 3(8 – 4x) < 6(x – 5), we need to solve it step by step:

Simplify the inequality:

24 - 12x < 6x - 30

Combine like terms:

-12x - 6x < -30 - 24

-18x < -54

Divide both sides of the inequality by -18, remembering to flip the inequality sign:

x > (-54) / (-18)

x > 3

The inequality tells us that x must be greater than 3. To represent this on a number line, we place an open circle at the value 3 and draw a bold line pointing to the right to indicate that the solution set includes all values greater than 3.

Therefore, option C accurately represent the solution set for the inequality 3(8 – 4x) < 6(x – 5).

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The value of m<RQS as required to be determined in the task content is; 70°.

It follows from the task content that the measure of angle RQS is to be determined as required.

Recall, the measure of the central angle subtended by an arc is twice that which it subtends at any point on the circumference.

Therefore, m<RPS = 2 • m<RQS.

m<RQS = 140°/2

m<RQS = 70°.

Ultimately, the measure of angle RQS are; 70°.

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Answer:

B , D , A

Step-by-step explanation:

(4x³ - 4 + 7x) - (2x³ - x - 8)

distribute the first parenthesis by 1 and the second by - 1

= 4x³ - 4 + 7x - 2x³ + x + 8 ← collect like terms

= 2x³ + 8x + 4 ← equivalent to expression B

---------------------------------------------------------------

(- 3x² + [tex]x^{4}[/tex] + x) + (2[tex]x^{4}[/tex] - 7 + 4x) ← remove parenthesis

= - 3x² + [tex]x^{4}[/tex] + x + 2[tex]x^{4}[/tex] - 7 + 4x ← collect like terms

= 3[tex]x^{4}[/tex] - 3x² + 5x - 7 ← equivalent to expression D

------------------------------------------------------------------

(x² - 2x)(2x + 3)

each term in the second factor is multiplied by each term in the first factor, that is

x²(2x + 3) - 2x(2x + 3) ← distribute parenthesis

= 2x³ + 3x² - 4x² - 6x ← collect like terms

= 2x³ - x² - 6x ← equivalent to expression A

Answer:

[tex]m = \frac{1 - ( - 7)}{9 - ( - 3)} = \frac{8}{12} = \frac{2}{3} [/tex]

B is the correct answer.

Answer:

a) x² - x - 12 = 0

Step-by-step explanation:

We have equation A = equation B

⇒ x² - 9 = x + 3

⇒ x² - 9 - x - 3 = 0

⇒ x² - x - 12 = 0

Approximately 2 tissues were used by each groom's guest at the wedding.

The calculation is as follows:

180 tissues ÷ 90 guests = 2 tissues per guest.

To determine how many tissues each groom's guest used, we need to find the average number of tissues per guest. We start by adding up the number of tissues used by the groom's guests, which is 180.

Then, we divide this total by the number of groom's guests, which is 90. This division gives us an average of 2 tissues per guest.

By dividing the total number of tissues used by the total number of guests, we can find the average number of tissues per guest. In this case, each groom's guest used approximately 2 tissues.

It's important to note that this calculation assumes an equal distribution of tissues among all the groom's guests.

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Answer:

3x - x²

Step-by-step explanation:

x(3 - x)

each term in the parenthesis is multiplied by the x outside. This is called the Distributive law

= (x × 3) + (x× - x)

= 3x + (- x²)

= 3x - x²

The answer is:

Work/explanation:

To simplify this expression, we will use the distributive property:

[tex]\sf{x(3-x)}[/tex]

Distribute the x:

[tex]\sf{x\cdot3-x\cdot x}[/tex]

[tex]\sf{3x-x^2}[/tex]

The area of quadrilateral QUAD is 2.5 square units.

To find the area of quadrilateral QUAD with vertices Q (-4, 3), U (3, 6), 1 (6, 3), and D (1, -4), we can use the Shoelace formula (also known as Gauss's area formula or the surveyor's formula).

The Shoelace formula states that the area of a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn) can be calculated as:

[tex]Area = 1/2 * |(x1y2 + x2y3 + ... + xny1) - (x2y1 + x3y2 + ... + x1yn)|[/tex]

Using this formula, we can calculate the area of quadrilateral QUAD as follows:

Area = [tex]1/2 * |(-46 + 33 + 6*(-4) + 13) - (33 + 6*(-4) + 1*(-4) + (-4)*3)|[/tex]

Simplifying the expression, we get:

[tex]Area = 1/2 * |(-24 + 9 - 24 + 3) - (9 - 24 - 4 - 12)|Area = 1/2 * |(-36) - (-31)|Area = 1/2 * |-36 + 31|Area = 1/2 * |-5|Area = 1/2 * 5Area = 5/2[/tex]

The area of quadrilateral QUAD is 2.5 square units.

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The area of quadrilateral QUAD is 17.5 square units.

The area of quadrilateral QUAD, we can use the Shoelace Formula, also known as the Gauss's Area Formula.

The formula states that if the coordinates of the vertices of a polygon are given in order, then the area of the polygon can be calculated using the following formula:

Area = 1/2 × |(x1y2 + x2y3 + ... + xn-1yn + xny1) - (y1x2 + y2x3 + ... + yn-1xn + ynx1)|

Let's apply this formula to find the area of quadrilateral QUAD:

Q (-4, 3)

U (3, 6)

A (6, 3)

D (1, -4)

Area = 1/2 × |(-4 × 6 + 3 × 3 + 6 × (-4) + 3 × (-1)) - (3 × 3 + 6 × (-4) + (-4) × (-1) + (-1) × (-4))|

Area = 1/2 × |(-24 + 9 - 24 - 3) - (9 - 24 + 4 + 4)|

Area = 1/2 × |(-42) - (-7)|

Area = 1/2 × |-42 + 7|

Area = 1/2 × |-35|

Area = 1/2 × 35

Area = 17.5

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Answer:

[tex]\text{a.} \quad m\angle NLM=93^{\circ}[/tex]

[tex]\text{c.} \quad m\angle FHG=31^{\circ}[/tex]

Step-by-step explanation:

The inscribed angle in the given circle is ∠NLM.

The intercepted arc in the given circle is arc NM = 186°.

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc.

Therefore:

[tex]m\angle NLM=\dfrac{1}{2}\overset{\frown}{NM}[/tex]

[tex]m\angle NLM=\dfrac{1}{2} \cdot 186^{\circ}[/tex]

[tex]\boxed{m\angle NLM=93^{\circ}}[/tex]

[tex]\hrulefill[/tex]

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc. Therefore:

[tex]m\angle HFG=\dfrac{1}{2}\overset{\frown}{HG}[/tex]

[tex]m\angle HFG=\dfrac{1}{2}\cdot 118^{\circ}[/tex]

[tex]m\angle HFG=59^{\circ}[/tex]

As line segment FH passes through the center of the circle, FH is the diameter of the circle. Since the angle at the circumference in a semicircle is a right angle, then:

[tex]m\angle FGH = 90^{\circ}[/tex]

The interior angles of a triangle sum to 180°. Therefore:

[tex]m\angle FHG + m\angle HFG + m\angle FGH =180^{\circ}[/tex]

[tex]m\angle FHG + 59^{\circ} + 90^{\circ} =180^{\circ}[/tex]

[tex]m\angle FHG +149^{\circ} =180^{\circ}[/tex]

[tex]\boxed{m\angle FHG =31^{\circ}}[/tex]

The amount of medication needed for 30 days is approximately 59.84 tablets of carprofen.

To calculate the amount of carprofen medication needed for 30 days, we need to consider the prescribed dosage, the weight of the patient, and the concentration of the tablets.

The prescribed dosage is 2.2 mg/kg BID x 30d. This means that the patient should take 2.2 milligrams of carprofen per kilogram of body weight, twice a day, for 30 days.

The weight of the patient is 34 kilograms. So, we need to calculate the total amount of carprofen needed for the entire treatment period.

First, we calculate the daily dosage by multiplying the weight of the patient (34 kg) by the prescribed dosage (2.2 mg/kg).

Daily dosage = 34 kg * 2.2 mg/kg = 74.8 mg/day.

Since the medication is prescribed twice a day, we multiply the daily dosage by 2 to get the total dosage per day.

Total dosage per day = 74.8 mg/day * 2 = 149.6 mg/day.

Finally, to find the total amount of medication needed for 30 days, we multiply the total dosage per day by the number of days.

Total medication needed = 149.6 mg/day * 30 days = 4488 mg.

Since the concentration of the tablets is 75 mg, we divide the total medication needed by the tablet concentration to find the number of tablets required.

Number of tablets needed = 4488 mg / 75 mg = 59.84.

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Answer:

We have the slope of the line, which is 2 and a point that is (1, -1).

To find the point-slope form of the line, we use the equation:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is the given point.

Substituting in the values we have, we get:

y - (-1) = 2(x - 1)

Simplifying this equation, we get:

y + 1 = 2(x - 1)

Therefore, the answer is option C: y + 1 - 2(x - 1).

Answer:

There were 126 student tickets sold and 252 adult ticket sold.

Step-by-step explanation:

Let x be the number of adult tickets sold

y be the number of students tickets sold

Twice as many student tickets as adult tickets were sold

a.

x = 2y  ---equation 1

6x + 4y = 2016   ---equation 2

b.

Substitute equation 1 to equation 2

6(2y) + 4y = 2016

12y + 4y = 2016

16y = 2016

Divide both sides of the equation by 16

16y/16 = 2016/16

y = 126

Substitute y = 126 to equation 1

x = 2y

x = 2(126)

x = 252

Answer:

8cm

Step-by-step explanation:

perimiter A = 11+11+4+4=30

perimiter B = 4+4+8+8=24

24+30=54

perimeter c =4+8+8+4+11+7+4=46

so perimiter of c is 8cm shorter than A and B total

hope this helps

The diagonal distance from home plate to third base is approximately √16,200 feet.

The correct answers are:

B. 90 feet

C. √16,200 feet

D. 180 feet.

In baseball, the bases are arranged in a square shape.

The distance between each base is 90 feet.

Therefore, the correct answer for the distance a player at first base has to throw to a player at third base is 90 feet (option B).

To find the diagonal distance from home plate to third base, we can use the Pythagorean theorem.

Since the area of the baseball field is a square, the diagonal distance represents the hypotenuse of a right triangles.

The two legs of the right triangle are the sides of the square, which are 90 feet each.

Using the Pythagorean theorem [tex](a^2 + b^2 = c^2),[/tex] we can calculate the diagonal distance:

a = b = 90 feet

[tex]c^2 = 90^2 + 90^2[/tex]

[tex]c^2 = 8,100 + 8,100[/tex]

[tex]c^2 = 16,200[/tex]

c = √16,200 feet (option C)

Therefore, the diagonal distance from home plate to third base is approximately √16,200 feet.

The options A, √180 feet, and 180 feet are incorrect because they do not represent the correct distances in the given scenario.

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The surface area and volume of the composite solid is are 1720ft² and 3563.33 ft³ respectively.

The area occupied by a three-dimensional object by its outer surface is called the surface area.

The surface area of the solid = lateral area of pyramid + surface area of cuboid

lateral area of pyramid = 4 × 1/2 bh

= 4 × 1/2 × 10× 12

= 120×2 = 240 ft²

Surface area of the cuboid = 2( 100+ 320+ 320)

= 2( 740)

= 1480 ft²

Surface area of the composite solid = 240 + 1480

= 1720 ft²

Volume of the composite solid = volume of cuboid + volume of pyramid

volume of cuboid = 10×10×32 = 3200ft²

volume of pyramid = 1/3base area × height

height of the pyramid is calculated as;

diagonal of base = √ 10²+10²

= √200

= 14.14

h² = 13²-7.07²

h² = 169 - 49.98

h² = 119.02

h = 10.9 ft

Volume of pyramid = 1/3 × 100 × 10.9

= 363.33 ft³

Volume of the composite solid = 3200+363.33

= 3563.33 ft³

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The correct answer is By finding the quotient of the bases to be one third and canceling common factors. Option D.

The Quotient of Powers Property states that when dividing two powers with the same base, you can subtract the exponents. In the given expression, we have three to the fourth power divided by nine.

To simplify this expression using the Quotient of Powers Property, we first need to recognize that nine can be written as three squared, since 3 multiplied by itself gives 9.

So, we have (3^4) / (3^2). According to the Quotient of Powers Property, we subtract the exponents: 4 - 2.

This gives us 3^(4-2), which simplifies to 3^2. Therefore, the expression three to the fourth power all over nine equals three squared.

It states that we find the quotient of the bases to be one third and cancel common factors. In this case, the bases are 3 and 3, and their quotient is indeed one third. Additionally, there are no common factors that can be canceled, as the expression does not contain any variables or additional terms.

Therefore, By finding the quotient of the bases to be one third and canceling common factors. accurately describes the steps involved in simplifying the expression using the Quotient of Powers Property.

We find the quotient of the bases (one third) and cancel common factors (which is not applicable in this case). Option D is correct.

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Note the complete question is

Explain how the Quotient of Powers Property was used to simplify this expression. (1 point)

Three to the fourth power all over nine equals three squared

A.) By simplifying 9 to 32 to make both powers base three and adding the exponents

B.) By simplifying 9 to 32 to make both powers base three and subtracting the exponents

C.) By finding the quotient of the bases to be one third and simplifying the expression

D.) By finding the quotient of the bases to be one third and cancelling common factors

The equation of the new function after shifting 6 units right and 4 units down is f(x) = (x + 6)² - 4.

If we are to apply the changes below to the quadratic parent function, F(x) = x², what is the equation of the new function, given that we are to shift 6 units to the right and 4 units down? We will approach this question by following the steps outlined below.

Step 1: Identify the parent function F(x) = x² and its transformations

Step 2: Write the equation of the new function

Step 3: Simplify the new equation of the function.Step 1: Identify the parent function F(x) = x² and its transformations

Here, we are given the quadratic parent function F(x) = x² and two transformations: shift 6 units right and shift 4 units down.

The general equation for the horizontal and vertical shifts of a quadratic function is given by:f(x) = a(x - h)² + k, where a, h, and k are constants.

The value of a determines the direction of opening of the parabola, while (h, k) represents the vertex of the parabola.

If a > 0, the parabola opens upwards, while a < 0, the parabola opens downwards. If the values of (h, k) are positive, the parabola is shifted right and up, respectively. On the other hand, if the values of (h, k) are negative, the parabola is shifted left and down, respectively.

Therefore, given the quadratic parent function F(x) = x² and two transformations: shift 6 units right and shift 4 units down, we can represent these changes by the following:

a = 1 (since the parabola opens upwards)h = -6 (since we are shifting the parabola 6 units to the right)k = -4 (since we are shifting the parabola 4 units down)

Step 2: Write the equation of the new function Now that we have identified the constants a, h, and k, we can write the equation of the new function as follows:f(x) = a(x - h)² + kf(x) = 1(x - (-6))² + (-4)Replacing the constants a, h, and k in the equation, we have:f(x) = (x + 6)² - 4

Step 3: Simplify the new equation of the function.f(x) = (x + 6)² - 4= (x + 6)(x + 6) - 4= x² + 12x + 36 - 4= x² + 12x + 32Therefore, the equation of the new function after shifting 6 units right and 4 units down is f(x) = x² + 12x + 32.

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Answer:

a. Let tomorrow's temperature be x

b.

[tex] |93 - x | < 10[/tex]

c.

[tex] - 10 < 93 - x < 10 \\ - 103 < - x < - 83 \\ 103 > x > 83[/tex]

Answer:

hope you understand it and please follow me

It will take approximately 5.33 minutes for Jeff to catch up to Roger. Both Jeff and Roger will have run a distance of 1 mile.

Let's calculate the time it takes for Jeff to catch up to Roger and the distance each will have run.

First, we need to determine how much distance Roger covers during the 1-minute head start. Since Roger runs one mile in 9 minutes, in 1 minute, he would cover 1/9 of a mile.

Now, let's set up an equation to represent the time it takes for Jeff to catch up to Roger:

Distance covered by Jeff = Distance covered by Roger + Distance covered during the head start

Let's denote the time it takes for Jeff to catch up as t minutes. During this time, Jeff runs a distance of 1 mile, while Roger runs a distance of 1/9 of a mile (from the head start).

Distance covered by Jeff = Distance covered by Roger + Distance covered during the head start

1 mile = (1/9) mile + (6 minutes) × t × (1 mile/6 minutes)

Now, let's solve this equation to find the time it takes for Jeff to catch up:

1 = (1/9) + (1/6)t

Multiplying the equation by the least common multiple (LCM) of 9 and 6, which is 18, to clear the fractions:

18 = 2 + 3t

Subtracting 2 from both sides:

16 = 3t

Dividing by 3:

t = 16/3 = 5.33 minutes (rounded to two decimal places)

Therefore, it will take approximately 5.33 minutes for Jeff to catch up to Roger. Both Jeff and Roger will have run a distance of 1 mile.

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If all the values in the series are the same, the A.M, G.M, and H.M are all equal and can be represented as A.M = G.M = H.M = x.

If all the values in the series are the same, the arithmetic mean (A.M), geometric mean (G.M), and harmonic mean (H.M) will all be equal.

Let's consider a series with the same value repeated n times, denoted as x:

Series: x, x, x, ..., x (n times)

Arithmetic Mean (A.M):

The arithmetic mean is calculated by summing all the values in the series and dividing by the total number of values. In this case, the sum of all the values is nx, and since there are n values, the arithmetic mean is (nx) / n = x. So, A.M = x.

Geometric Mean (G.M):

The geometric mean is calculated by taking the nth root of the product of all the values in the series. In this case, the product of all the values is x^n, and since there are n values, the nth root of x^n is x. So, G.M = x.

Harmonic Mean (H.M):

The harmonic mean is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of all the values in the series. Since all the values are the same, the reciprocal of each value is 1/x. The arithmetic mean of the reciprocals is (1/x + 1/x + ... + 1/x) / n = (n/x) / n = 1/x. Taking the reciprocal of 1/x gives x. So, H.M = x.

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The values x = -5 and x = 3 make the second expression undefined. The correct answers are:

x = -5

x = 3

x= -3

To determine the values for which the given expressions are undefined, we need to find the values that make the denominators equal to zero.

First expression: [tex]\frac{3x}{(x^2 - 9)}[/tex]

For this expression, the denominator is (x^2 - 9). It will be undefined when the denominator equals zero:

x^2 - 9 = 0

Factoring the equation, we have:

(x - 3)(x + 3) = 0

Setting each factor equal to zero, we get:

x - 3 = 0 --> x = 3

x + 3 = 0 --> x = -3

So, the values x = 3 and x = -3 make the first expression undefined.

Second expression: [tex]\frac{(x + 4)}{(x^2 + 2x - 15)}[/tex]

For this expression, the denominator is (x^2 + 2x - 15). It will be undefined when the denominator equals zero:

x^2 + 2x - 15 = 0

Factoring the equation, we have:

(x + 5)(x - 3) = 0

Setting each factor equal to zero, we get:

x + 5 = 0 --> x = -5

x - 3 = 0 --> x = 3

So, The second expression is ambiguous because x = -5 and x = 3.

Consequently, the right responses are x = -5, x = 3 and x= -3.

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The measure of the outside angle F indicated in the figure is 61 degrees,

The external angle theorem states that "the measure of an angle formed by two secant lines, two tangent lines, or a secant line and a tangent line from a point outside the circle is half the difference of the measures of the intercepted arcs.

Expressed as:

Outside angle = 1/2 × ( major arc - minor arc )

From the figure:

Major arc = 195 degrees

Minor arc = 73 degrees

Outside angle F = ?

Plug the value of the minor and major arc into the above formula and solve for the outside angle F:

Outside angle = 1/2 × ( major arc - minor arc )

Outside angle = 1/2 × ( 195 - 73 )

Outside angle = 1/2 × ( 122 )

Outside angle = 122/2

Outside angle = 61°

Therefore, the outside angle measures 61 degrees.

Option C) 61° is the correct answer.

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