In this activity, you’ll use the inspection method to rewrite a rational expression, a(x)/b(x), in the form q(x) + r(x)/b(x).Answer these questions to step through the process of rewriting x^2-5x+7/x-9Part ACan the polynomial in the numerator of the expression x^2-5x+7/x-9 be factored to derive (x-9) as a factor?Answer is noPart DWhat number must be added to the numerator to get the new constant term you identified in Part C?Part EAdd the number you calculated in part D to the numerator, and then subtract the number to keep the value of the expression unchanged.Part F Rewrite the numerator so it contains a trinomial that can be faced with x-9 as a common factor, and then write it in the factored formPart GRewrite the expression you found in part F as the sum of two rational expressions with (x-9) as their common denominator Part HReduce the first fraction and write the expression in this format:A(x)/b(x) = q(x)+ r(x)/b(x)

In This Activity, Youll Use The Inspection Method To Rewrite A Rational Expression, A(x)/b(x), In The

Answers

Answer 1

The expression is:

[tex]\frac{x^2-5x+7}{x-9}[/tex]

Part B

To get -9 to -5, we need to add 4. This is important because the factored form will be something like this:

[tex]x^2-5x+7=(x-9)(x+a)[/tex]

And when we distribute it, the middle term will be the sum of -9 and a, so we if we want it to be -5 (as the given expression) a has to be 4.

Part C

Now, looking to the constant part, it will be the multiplication of -9 and a, since we know that a is 4, the constant term is:

[tex]-9\cdot4=-36[/tex]

So, we need a constant term of -36 in the numerator.

Part D

Since we already got 7 in the numerator, we have to add -43 to get it to -36.

Part E

[tex]\frac{x^2-5x+7}{x-9}=\frac{x^2-5x+7+(-43)-(-43)}{x-9}=\frac{x^2-5x-36+43}{x-9}[/tex]

Part F

[tex]\frac{x^2-5x+-36+43}{x-9}=\frac{(x-9)(x+4)+43}{x-9}[/tex]

Part G

[tex]\frac{(x-9)(x+4)+43}{x-9}=\frac{(x-9)(x+4)}{x-9}+\frac{43}{x-9}[/tex]

Part H

[tex]\frac{(x-9)(x+4)}{x-9}+\frac{43}{x-9}=x+4+\frac{43}{x-9}[/tex]

So:

[tex]\frac{x^2-5x+7}{x-9}=x+4+\frac{43}{x-9}[/tex]


Related Questions

(c) In the diagram below:ARga nainP.50°B65%DNot drawn to scale(i) Calculate the angle BDC (ii) Calculate angle ABD (iii) Find angle BAD(iv) What type of triangle is triangle ABD ?CS

Answers

Given: Parallel lines PQ and RS. Triangle ABD and BDC are such that

[tex]\begin{gathered} BD=CD \\ m\angle ABR=50\degree \\ m\angle ADB=65\degree \end{gathered}[/tex]

Required: To determine the triangle ABD type and calculate the angle BDC, ABD, and angle BAD.

Explanation: Since line PQ is parallel to line RS,

[tex]\angle ADB=\angle DBC=65\degree[/tex]

Now since BD=CD, triangle BCD is an isosceles triangle. Hence,

[tex]\angle DBC=\angle DCB=65\degree[/tex]

Now, in triangle BCD, we have

[tex]\begin{gathered} \angle B+\angle C+\angle D=180\degree\text{ \lparen Angle sum property\rparen} \\ 65\degree+65\degree+\angle D=180\degree \\ \angle D=50\degree \end{gathered}[/tex]

Now RS is a straight line. Hence at point B, we have

[tex]\begin{gathered} 50\degree+\angle ABD+\angle DBC=180\degree\text{ \lparen Linear pair\rparen} \\ \angle ABD=65\degree \end{gathered}[/tex]

Finally, in triangle ABD, we have

[tex]\begin{gathered} \angle A+\angle B+\angle D=180\degree \\ \angle A+65\degree+65\degree=180\degree \\ \angle A=50\degree \end{gathered}[/tex]

Now since in triangle ABD, we have

[tex]\angle ABD=\angle ADB[/tex]

The triangle ABD is isosceles.

Final Answer:

[tex]\begin{gathered} \angle BDC=50\degree \\ \angle ABD=65\degree \\ \angle BAD=50\degree \end{gathered}[/tex]

The triangle ABD is isosceles.

Can you please help me out with a question

Answers

Answer:

3 degrees

Explanation:

Using the theorem that states that the measure of the angle at the circumference is equal to the half of its intercepted arc. Hence;

31x+ 3 = 1/2(192)

31x + 3 = 96

Subtract 3 from both sides

31x + 3 - 3 = 96 - 3

31x = 93

Divide both sides by 31

31x/31 = 93/31

x = 3

Hence the value of x is 3 degrees

Will a truck that is 14 feet wide carrying a load that reaches 12 feet above the ground clear the semielliptical arch on the one-way road that passes under the bridge shown in the figure on the right?

Answers

Given the equation of an elipse

[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=1[/tex]

from the question,

[tex]\begin{gathered} \text{major axis}\Rightarrow2a \\ \therefore2a=52 \\ a=\frac{52}{2}=26ft \\ b=13ft \end{gathered}[/tex]

Given that

[tex]x=14ft[/tex]

Substitute, for a,b, and x in the elipse formula to find y

[tex]\begin{gathered} \frac{14^2}{26^2}+\frac{y^2}{13^2}=1 \\ \frac{196}{676}+\frac{y^2}{169}=1 \end{gathered}[/tex]

Multiply through by 169

[tex]\begin{gathered} 49+y^2=169 \\ y^2=169-49 \\ y^2=120 \\ y=\sqrt[]{120}=10.95ft \end{gathered}[/tex]

Hence, it clear the arch because the height of the archway of the bridge 7 feet from the center is approximatelyfeet

Find anangle 0 coterminal to -560°, where 0° < 0 < 360°.

Answers

Given the angle -560

The coterminal angle will be:

[tex]\theta=-560+360=-200+360=160[/tex]

So, the answer will be 160

Round 14.235 to the nearest tenth, hundredth, one and ten

Answers

The number 14.235 would round down to 14.2

What is rounding a number to some specific place?

Rounding some number to a specific value is making its value simpler (therefore losing accuracy), mostly done for better readability or accessibility.

Rounding to some place keeps it accurate on the left side of that place but rounded or sort of like trimmed from the right in terms of exact digits.

We need to round  14.235 to the nearest tenth, hundredth, one and ten

We can see that the 35 is below 50 so it goes down, and it rounds down to 14.2 instead of, 14.62 then that would round up to the decimal is higher than 50.

Since it is 235, then it rounds down to 14.2

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All of the following are equivalent exceptO (4)(y)O 4+ y04.1O 4 yASK FOR HELPUNT QUESTION

Answers

[tex]\begin{gathered} 4y \\ (4)(y)=4y \\ 4.y=4y \\ 4+y\ne4y \\ So\text{ all are eqivalent except (4+y)} \end{gathered}[/tex]

the Firgure shows two triangles on a coordinate grid: Which set of transformations have been performed on triangle ABC to form triangle A'B'C'? A) Dilation by a scale factor of 1/3 followed by reflection about the x-axisB) Dilation by a scale factor of 3 followed by reflection about the x-axis C) Dilation by a scale factor of 1/3 followed by reflection about the y-axis D) Dilation by a scale factor of 3 followed by reflection about the y-axis

Answers

In transformations, the Pre-Image is the original figure and the Image is the figure transformated.

In this case you can identify that the Pre-Image is the triangle ABC and the Image is the triangle A'B'C'.

Notice that the vertices of ABC are:

[tex]A(-3,3);B(0,0);C(-6,-3)[/tex]

By definition, when the scale factor used in the dilation is between 0 and 1, the Image obtained is a reduction and, therefore, it is smaller than the Pre-Image. Since A'B'C' is smaller than ABC, then you can determine that ABC was dilated by this scale factor:

[tex]sf=\frac{1}{3}[/tex]

When a figure is reflected across the y-axis, the rule is:

[tex](x,y)\rightarrow\mleft(-x,y\mright)[/tex]

If you dilate ABC by the scale factor shown above, and then you reflect it across the y-axis, the coordinates of the Image will be:

[tex]\begin{gathered} A\mleft(-3,3\mright)\rightarrow A^{\prime}(-(\frac{-3}{3}),\frac{3}{3})\rightarrow A^{\prime}(1,1) \\ \\ B\mleft(0,0\mright)\rightarrow B^{\prime}\mleft(0,0\mright) \\ \\ C\mleft(-6,-3\mright)\rightarrow C^{\prime}(-(\frac{-6}{3}),\frac{-3}{3})\rightarrow C^{\prime}(2,-1) \end{gathered}[/tex]

Notice that the coordinates of A'B'C' shown in the picture match with the vertices found above.

Therefore, the answer is: Option C.

i would like help understanding this form of math please.

Answers

Question:

Solution:

If we have the formula:

[tex]\text{Height = }\frac{Cons\tan t}{\text{Width}}[/tex]

14. In your rectangular backyard, you knowthe width of the yard is three lessthan four times the length. If the perimeterof your yard is 24 yards, what isthe width?18 3/5yards3 yards9 yards15 yards

Answers

ANSWER:

3rd option: 9 yards

STEP-BY-STEP EXPLANATION:

Given that:

Length = L

Width = W = 4L - 3

The perimeter is the sum of all the sides, therefore:

[tex]\begin{gathered} p=L+L+W+W \\ \\ \text{ We replacing:} \\ \\ 24=L+L+4L-3+4L-3 \\ \\ \text{ We solve for L} \\ \\ 24+3+3=10L \\ \\ L=\frac{30}{10} \\ \\ L=3\text{ yd} \\ \\ \text{ Therefore:} \\ \\ W=4L-3=4(3)-3=9\text{ yd} \end{gathered}[/tex]

So the correct answer is 3rd option: 9 yards

4+(6x2²)-9 use pemdas

Answers

Given:

[tex]4+(6\times2^2)-9[/tex]

Required:

To solve the given expression.

Explanation:

Consider

[tex]\begin{gathered} =4+(6\times2^2)-9 \\ \\ =4+(6\times4)-9 \\ \\ =4+24-9 \\ \\ =28-9 \\ \\ =19 \end{gathered}[/tex]

Final Answer:

[tex]4+(6\times2^2)-9=19[/tex]

ginny is raising pumpkins to enter a contest to see who can grow the heaviest pumpkin. her best pumpkin weighs 22 pounds and is growing at the rate of 2.5 pounds per week. martha planted her pumpkins late. her best pumpkin weighs 10 pounds but she expects it grow 4 pounds per week. define the "let" statements for x and y. then write equations that represent the weight of ginny and martha's pumpkins.Let x=Let y=ginny's equations=Martha's equation:

Answers

Let x=

1) Gathering the data

Ginny

Best pumpkin: 22 pounds

The growing rate of 2.5 pounds per week

Martha

Best pumpkin: 10 pounds

The growing rate: 4 pounds per week

Let x for the growing rate and y for the weight

2) Setting equations

Ginny's equation

2.5x=22

Martha's equation: ​

4x=10

The coordinates of the focus are (2,-7/4), the coordinates of the endpoints of the latus rectum are (3/2,-7/4) and (5/2,-7/4). The equation of the directions is y=-9/4, and the equation of the axis of symmetry is x=2.

Answers

General equation of a parabola:

[tex](x-h)^2=4p(y-k)[/tex]

Equation of the axis of symmetry:

x = h

In this case, the axis of symmetry is x = 2, then h = 2.

Equation of the directrix:

y = k - p

In this case, the equation of the directrix is y = -9/4, then:

-9/4 = k - p (eq. 1)

Equation of the focus:

F(h, k+p)

In this case, the coordinates of the focus are (2,-7/4), then:

-7/4 = k + p (eq. 2)

Adding equation 1 to equation 2:

-9/4 = k - p

+

-7/4 = k + p

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

-4 = 2k

(-4)/2 = k

-2 = k

Substituting this result into equation 2 and solving for p:

-7/4 = -2 + p

-7/4 + 2 = p

1/4 = p

Substituting with h = 2, k = -2, and p = 1/4 into the general equation, we get:

[tex]\begin{gathered} (x-2)^2=4\cdot\frac{1}{4}(y-(-2)) \\ (x-2)^2=y+2 \end{gathered}[/tex]

find the values of x and y

Answers

From the given figure

Since every two opposite sides are parallel

AB // DC

AD // BC

Then the given quadrilateral is a parallelogram

ABCD is a parallelogram

From the properties of the parallelogram,

Every 2 opposite sides are equal in length, then

AB = DC

AB = x + 2, and DC = 13, then

x + 2 = 13

Subtract 2 from both sides to find x

x + 2 - 2 = 13 - 2

x = 11

Since the opposite angles in the parallelogram are equal in measures

Since

m

Since m

y = 70

The value of x is 11 and the value of y is 70

The length of the rectangle below is 7 than it’s width. Given that the total distance around the rim of the shape is 46 units, what is the value of x?

Answers

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Write the equation of the line parallel to Y equals 2/3X +1 through the point (0,-4)  use slope intercept form

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Writing the slope-intercept form of a linear equation, we have:

[tex]y=mx+b[/tex]

Where m is the slope and b is the y-intercept.

Since parallel lines have the same slope, we can see that the slope of the line y = 2/3x + 1 is equal m = 2/3, so for our equation we also have m = 2/3.

Now, using the point (0, -4), we have:

[tex]\begin{gathered} y=\frac{2}{3}x+b \\ (0,-4)\colon \\ -4=\frac{2}{3}\cdot0+b \\ b+0=-4 \\ b=-4 \end{gathered}[/tex]

So our equation is:

[tex]y=\frac{2}{3}x-4[/tex]

y = 2/3x - 4

Devonte creates a scatter plot of the relationship between his hourly pay in dollars, y, and the number of customers he serves at a coffee shop, X. He calculates the equation of the trend line to be y = 2.52 +7. Part A What does the y-intercept represent? Enter the correct answers in the boxes. per hour when he serves customers. The y-intercept represents that Devonte earns $

Answers

Given equation of line is,

The following is a sample of 20 measurements.Answer b part

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b)

Given:

[tex]\begin{gathered} \bar{x}=10.2 \\ s=2.12 \end{gathered}[/tex]

Hence,

[tex]\begin{gathered} \bar{x}\pm s=10.2\pm2.12 \\ \bar{x}+s=12.32 \\ \bar{x}-s=8.08 \end{gathered}[/tex]

So, the measurements in the data between 8.08 and 12.32 are 11, 9, 12, 10 12, 12 , 12, 9, 9, 9, 11, 11, 12 and 11.

Therefore, the number of measurements in interval x±s is 14.

The percentage of the measurements that fall between the interval x±s is,

[tex]\text{Percent}=\frac{14}{20}\times100=70[/tex]

Therefore, the percentage of the measurements that fall between the interval x±s is 70%.

Now,

[tex]\begin{gathered} \bar{x}\pm2s=10.2\pm2\times2.12 \\ \bar{x}\pm2s=10.2\pm4.24 \\ \bar{x}+2s=14.44 \\ \bar{x}-2s=5.96 \end{gathered}[/tex]

So, all the measurements in the data are between 5.96 and 14.44.Therefore, the number of measurements in interval x±2s is 20.

Therefore, the percentage of the measurements that fall between the interval x±2s is 100%.

Now,

[tex]\begin{gathered} \bar{x}\pm3s=10.2\pm3\times2.12 \\ \bar{x}\pm3s=10.2\pm6.36 \\ \bar{x}+3s=16.56 \\ \bar{x}-3s=3.84 \end{gathered}[/tex]

So, all the measurements in the data are between 3.84 and 16.56.Therefore, the number of measurements in interval x±3s is 20.

Therefore, the percentage of the measurements that fall between the interval x±3s is 100%.

Last part: compare the percentage .

According to empirical rule, approximately 68% of the measurements in a sample will fall within the interval x±s.

From part b, the obtained percentage of measurements that fall within the interval x±s is 70%.

Therefore, percentage of measurements that fall within the interval x±s is greater than the predicted percentage for x±s using the empirical rule.

Option C is correct.

Error Analysis Denzel identified (3, 2) as a point on the line y - 2 = 2/3 (x + 3). What is the error that Denzel made?

Answers

Slope point formula:

y-y1= m (x-x1)

Where:

m= slope

(y1,x1) = point of the line

For:

y - 2 = 2/3 (x + 3)

m= 2/3

y1= 2

x1= -3

The error is that the point is not (3,2) is (-3,2)

y-2 = 2/3 (x-(-3))

y-2 = 2/3 (x+3)

2.3= p + 0.6What does p equal?

Answers

The given equation is

[tex]2.3=p+0.6[/tex]

First, we subtract 0.6 on each side

[tex]\begin{gathered} 2.3-0.6=p+0.6-0.6 \\ 1.7=p \end{gathered}[/tex]Therefore, p is equal to 1.7.

Please help us in figuring out this math problem so that we can move onto the next one thank you very much

Answers

All numbers in scientific notation or standard form are written in the form

[tex]m\cdot10^n[/tex]

where m is a number between 1 and 10 and the exponent n is a positive or negative integer.

To convert 64500 into scientific notation, follow these steps:

1. Move the decimal 4 times to left in the number so that the resulting number, m = 6.45, is greater than or equal to 1 but less than 10

2. Since we moved the decimal to the left the exponent n is positive

n = 4

3. Write in the scientific notation form, m × 10^n

= 6.45 × 10^4

In ΔTUV, t = 82 inches, v = 86 inches and ∠V=41°. Find all possible values of ∠T, to the nearest degree.

Answers

The value of ∠T is 38.722° as the definition of angle is "An angle is created by joining two line segments at one point, or we can say that an angle is the combination of two line segments at a common endpoint".

What is angle?

An angle is created by joining two line segments at one point, or we can say that an angle is the combination of two line segments at a common endpoint. When two straight lines or rays intersect at a single endpoint, an angle is created. The vertex of an angle is the location where two points come together. The Latin word "angulus," which means "corner," is where the word "angle" originates. Based on measurement, there are different kinds of angles in geometry. The names of fundamental angles include acute, obtuse, right, straight, reflex, and full rotation. A geometrical shape called an angle is created by joining two rays at their termini. In most cases, an angle is expressed in degrees.

Here,

Side t = 82

Side u = 128.98238

Side v = 86

Angle ∠T = 38.722°

Angle ∠U = 100.278°

Angle ∠V = 41°

∠T = sin⁻¹(t·sin(V)/v)

=38.722°

Since the definition of an angle is "An angle is created by joining two line segments at one point, or we can say that an angle is the combination of two line segments at a common endpoint," the value of ∠T is 38.722°.

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A total of 5000 tickets were sold for a raffle. the prizes are $1000, $500, $200, and $100. what price should be charged so there is a 60% profit per ticket?

Answers

Answer: $0.576

Step-by-step explanation:

The total amount in prizes is $1800.

For there to be 60% profit, the total cost of the tickets need to be [tex]1800(1.6)=\$ 2880[/tex].

Thus, each ticket must sell for [tex]\frac{2880}{5000}=\$ 0.576[/tex]

$0.576 should be charged so there is a 60% profit per ticket.

What is Unitary Method?

The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.

For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.

12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.

As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.

This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.

Given:

The prizes are $1000, $500, $200, and $100.

So, total prize = 1000+ 500+ 200+ 100 = $1800.

The, the price of ticket to break

= 1800 / 5000

= $0.36

Now, the price for 60% ticket = 0.36 (1 + 0.6)

                                                 = 0.36 x 1.6

                                                 = $0.576

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What is the center and the radius of the circle: ( x + 7 ) 2 + ( y - 1 ) 2 = 9 ?

Answers

Given:

There is a equation of circle given in the question as below

[tex]\left(x+7\right)^2+(y-1)^2=9[/tex]

Required:

We want to find the center and radius of given circle

Explanation:

The general equation of circle is

[tex](x-h)^2+(y-k)^2=r^2[/tex]

where (h,k) is the center of circle and r be the radius of circle

Now by comparing we get

[tex]\begin{gathered} (h,k)=(-7,1) \\ r^2=9\Rightarrow r=3 \end{gathered}[/tex]

Final answer:

C

I don't understand if this equation is a linear equation or not. Can you please help me?

Answers

we have the equation

[tex]\frac{x}{4}-\frac{y}{3}=1[/tex]

To remove the fractions, multiply both sides by (4*3=12)

[tex]\begin{gathered} \frac{12x}{4}-\frac{12y}{3}=12 \\ 3x-4y=12 \\ 4y=3x-12 \\ y=\frac{3}{4}x-3 \end{gathered}[/tex]

this is the equation of a line

that means

is a linear equation

30 points helps What are the coordinates of each vertex if the figure is rotated 180° clockwise about the origin?[G.CO.2, G.CO.4, G.C0.5]

Answers

The coordinates of each of the vertices after the figure is rotated 180° clockwise about the origin: A' = (2, -2), B' = (-2, -5), C' = (-6, -3), D' = (-4, 3)

Explanation:

The first thing we do is to write the vertices of the original shape:

A = (-2, 2)

B = (2, 5)

C = (6, 3)

D = (4, -3)

A rotation of (x, y) 180° clockwise about the origin = (-x, -y)

We take the negative of the x and y coordinates of the original shape

180° clockwise about the origin becomes:

A' = (-(-2), -2) = (2, -2)

B' = (-2, -5)

C' = (-6, -3)

D' = (-4, -(-3)) = (-4, 3)

The coordinates of each of the vertices after the figure is rotated 180° clockwise about the origin: A' = (2, -2), B' = (-2, -5), C' = (-6, -3), D' = (-4, 3)

Hello I need help with this here please, I was studying but I can’t get this

Answers

ANSWER

B. False

EXPLANATION

The Pythagorean Theorem states that the sum of the squares of the two legs of a right triangle, a and b, is equal to the square of the hypotenuse, c:

[tex]c^2=a^2+b^2[/tex]

And this theorem is true for all right triangles.

Hence, this statement is false.

drawing a sketch, giving an example, or providing a written description, please indicate themeaning of each of the following shapes.

Answers

For the given shapes, we will draw a sketch

a) A cone

the sketch of the cone will be as follows:

The cone has a circular base of radius = r, and a height of (h) and has a flat surface and curved surface as shown.

b) The diameter of the circle:

The diameter is a line segment (d) that connects two points lying on the circle through the center of the circle

c) The radius of the circle:

The radius of the circle (r) is a line segment that connects the center of the circle and any point lying on the circle

Which equation is an identity?O 3(x - 1) = x + 2(x + 1) + 1Ox-4(x + 1) = -3(x + 1) + 1O 2x + 3 = 1 (4x + 2) + 2(6x - 3) = 3(x + 1) – x-2

Answers

Identity equations are always true, no matter the values that the variables take.

We have to calculate for each one, and if the result gives a true statement, then the equation is an identity:

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

[tex]\begin{gathered} 3\left(x-1\right)=x+2\left(x+1\right)+1 \\ 3x-3=x+2x+2+1 \\ 3x-3=3x+3 \\ 3x-3x=3+3 \\ 0=6 \end{gathered}[/tex]

This is FALSE (for any value of x), so the equation is not an identity.

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

[tex]\begin{gathered} x-4\left(x+1\right)=-3\left(x+1\right)+1 \\ x-4x-4=-3x-3+1 \\ x(1-4+3)=-2+4 \\ 0=2 \end{gathered}[/tex]

This is FALSE, so the equation is not an identity.

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

[tex]\begin{gathered} 2x+3=14x+2+2 \\ 3-2-2=14x-2x \\ -1=12x \\ x=\frac{-1}{12} \end{gathered}[/tex]

This equation holds true only for x=-1/12, so it is not an identity.

4) (6x - 3) = 3(x + 1) – x-2



[tex]\begin{gathered} \left(6x-3\right)=3\left(x+1\right)-x-2 \\ 6x-3=3x+3-x-2 \\ 6x-3=2x+1 \\ 6x-2x=1+3 \\ 4x=4 \\ x=1 \end{gathered}[/tex]

This equation holds true only for x=1, so it is not an identity.

Neither of the options is an identity.

I don't understand how to add and subtract Intregers

Answers

Explanation

First of, you should know that integers are whole numbers.

There are positive integers (positive whole numbers, that is, normal whole numbers greater than 0, for example, 7, 98, 14 etc.) and there are negative integers (negative whole numbers, that is, whole numbers less than 0, for example, -3, -37, -101 etc.)

So, the first tip about adding and subtracting these numbers is to look at them in monetary terms.

Always look at positive numbers like money you have in your pockets (cash at hand).

And look at negative numbers like money you're owing someone.

So, we can then go through the different types of addition and subtraction of integers now.

- Addition of two positive numbers

** 2 + 2

You can interprete this simple addition as having $2 and another $2 is given to you, this means you've got $4 now.

2 + 2 = 4

** 17 + 7

You can interprete this simple addition as having $17 and another $7 is given to you, this means you've got $24 now.

17 + 7 = 24

- Subtraction of two positive integers

** 7 - 3

Look at this like having $7, then -3 means $3 is taken away from it, then you've got only $4 left.

7 - 3 = 4

** -15 + 10

This means you're owing $15, and you've got only $10, after paying the $10, there's still a debt of $5 left. So,

-15 + 10 = -5

Before the next two further types of adding/subtracting integers, weneed to also note the following

(+) × (+) = (+)

(+) × (-) = (-)

(-) × (+) = (-)

(-) × (-) = (+)

These helps us to simplify these additions and subtractions that involve a mix of positve numbers and negative numbers or just strictly working with negative numbers.

Addition of two negative numbers

** -5 + (-3)

Normally, with the former approach, this just means a debt of $5 is added to a debt of $3, these come together to give a bigger debt of $8.

But we can simplify the given equation further because we know that

(+) × (-) = (-), So,

-5 + (-3) = -5 - 3 (The plus sign before the -3 and the minus sign in the bracket multiply to give a negative/minus sign).

So,

-5 + (-3) = -5 -3 = -8

** -7 + (-4)

-7 + (-4) = -7 - 4 = -11

Subtraction of two negative integers

** -5 - (-5)

Recall that

(-) × (-) = (+), So,

-5 - (-5) = -5 + 5

Which then translates to owing $

It is 6 miles in a kayak to the Fish Islands from my house. The trip to the island takes 2 hourstraveling against the current and 1¼ hours for the return trip (with the current). How fast can Ipaddle the Kayak if there was no current? The answer can be rounded to the nearest tenth.Solve Algebraically using linear systems

Answers

It is given that the distance is 6 miles and the time is 2 hours upstream and one and a quarter hour downstream.

The time downstream is given by:

[tex]1\frac{1}{4}=\frac{4+1}{4}=\frac{5}{4}\text{ hours}[/tex]

Since the distance is constant, it follows:

[tex]\begin{gathered} \text{ Speed=}\frac{\text{ Distance}}{\text{ Time}} \\ \text{ Distance=SpeedxTime} \end{gathered}[/tex]

So the distance is constant hence:

[tex]\text{ Distance upstream=Distance Downstream}[/tex]

Let the speed of kayak be x and speed of current be y so the speed downstream is x+y and speed upstream is x-y so it follows:

[tex]\begin{gathered} \frac{5}{4}(x+y)=2(x-y) \\ 4\times\frac{5}{4}(x+y)=4\times2(x-y) \\ 5x+5y=8(x-y) \\ 5x+5y=8x-8y \\ 5y+8y=8x-5x \\ 13y=3x \\ x=\frac{13}{3}y \end{gathered}[/tex]

Use the individual equation to find x and y as follows:

[tex]\begin{gathered} 6=2(x-y) \\ 6=2(\frac{13}{3}y-y) \\ 3=\frac{13-3}{3}y \\ \frac{9}{10}=y \end{gathered}[/tex]

Hence the speed of the water current is 9/10 miles per hour.

The speed of the kayak is given by:

[tex]\begin{gathered} x=\frac{13}{3}y \\ x=\frac{13}{3}\times\frac{9}{10} \\ x=\frac{39}{10}=3.9\text{ miles per hour} \end{gathered}[/tex]

Hence the speed of the kayak without the water current is 3.9 miles per hour.

The time required without water current is:

[tex]\begin{gathered} \text{Time}=\frac{Dis\tan ce}{Speed} \\ t=\frac{6}{3.9}\approx1.5\text{ hours} \end{gathered}[/tex]

Hence it will take approximately 1.5 hours without the current.

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