What is the value of –2x2 + 4y if x = –2 and y = 3?

Answer:

x = 2.75+√30.5625

∠3 = ∠5 ≈ 113.923°

Step-by-step explanation:

We are given that ∠3 = (x+1)(x+4) and ∠5 = 16(x+3)-(x²-2) are corresponding angles, hence equal. We can equate the two angle expressions and solve the resulting quadratic for x.

... (x+1)(x+4) = 16(x+3)-(x²-2)

... x² +5x +4 -16x -48 +x² -2 = 0 . . . . . subtract the right side, eliminate parentheses

... 2x² -11x -46 = 0 . . . . . . . . . . . . . . . . . collect terms

Using the quadratic formula, we want to find

... x = (-b±√(b²-4ac))/(2a) . . . . for a=2, b=-11, c=-46

... x = (11 ±√((-11)² -4(2)(-46)))/(2(2)) = (11 ±√489)/4 = 2.75 ± √30.5625

The negative solution results in negative values for the angles, so only the positive solution is useful for this problem.

... x = 2.75+√30.5625 ≈ 8.27834

Using this value for x in either expression for the angle value, we get

... ∠3 = ∠5 = (8.27834+1)(8.27834+4) ≈ 113.923 . . . degrees

_____

It seems a little odd that this problem should result in irrational values for the variables. If we take ∠3 and ∠5 to be a linear pair, then the solution is x=6 and the angle measures are 70° and 110°. The solution is done basically the same way, except that you use the equation

... ∠3 + ∠5 = 180

and substitute the given expressions. The x² terms will cancel, leaving a linear equation easily solved.

(Since this is not the problem described here, the detailed working is left to the reader.)

see explanation below

(1) [tex]\frac{1}{5}[/tex] × [tex]\frac{2}{2}[/tex] = [tex]\frac{2}{10}[/tex] = 0.2

(2) [tex]\frac{6}{25}[/tex] × [tex]\frac{4}{4}[/tex] = [tex]\frac{24}{100}[/tex] = 0.24

(3) 2 [tex]\frac{3}{4}[/tex] = 2 +[tex]\frac{75}{100}[/tex] = 2.75

(4) 3 [tex]\frac{9}{10}[/tex] = 3 + 0.9 = 3.9

(5) 1.25 = 1 [tex]\frac{1}{4}[/tex] = [tex]\frac{5}{4}[/tex]

(6) 3.29 = 3 [tex]\frac{29}{100}[/tex] = [tex]\frac{329}{100}[/tex]

(7) 0.65 = [tex]\frac{65}{100}[/tex] = [tex]\frac{13}{20}[/tex] in simplest form

(8) 5.6 = 5 [tex]\frac{6}{10}[/tex] = 5 [tex]\frac{3}{5}[/tex] = [tex]\frac{28}{5}[/tex]

(9) he is incorrect

[tex]\frac{3}{5}[/tex] × [tex]\frac{20}{20}[/tex] = [tex]\frac{60}{100}[/tex] = 0.6 ≠ 3.5

Answer: These are some points of the grahp:

(-2,4)

(0, 3)

(2, 2)

Explanation:

1) f(x) =  -0.5x + 3, is the equation of the form y = mx + b

2) y = mx + b is slope-intercept  equation of a line where the slope is m and the y-intercept is b, so, f(x) = - 0.5x + b has slope m = -0.5 and y-intercept b = 3.

3) To graph f(x) = -0.5x + 3, follow these steps:

        x       f(x) = - 0.5x + 3

        -2         4

        0          3

        2          2

        4           1

        6          0

4) You can see the graph in the figure attached, and select any of the points on the line either by using the table or by using the equation f(x) = -0.5x + 3.

Based on the above, by the use the line tool,  the points of the graph will have the points of:

To graph the linear equation f(x) = -0.5x + 3, use the following steps:

Begin with a coordinate plane. Select two points that lie on the line. For example, you can choose x = 0 and x = 6. Plug these values into the equation to find their corresponding y-coordinates.

When x = 0, y = -0.5(0) + 3 = 3. So, one point is (0, 3).

When x = 6, y = -0.5(6) + 3 = 0. So, second  point is (6, 0).

Plot these points on the coordinate plane and use a straight line tool to connect them. This line represents the graph of f(x) = -0.5x + 3.

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Answer:

it is decreasing in the time interval 0<x<2 hours.

Step-by-step explanation:

Graph of first line going through ordered pairs A(0, 40) and B(2, 10).

Graph of second line going through ordered pairs B(2, 10) and C(4, 10). Graph of third line going through ordered pairs C(4, 10) and D(7, 50).

When movement is from point A to B, change in x = 2 and change in y = -30

Hence change for unit x = -30/2 = -15 ... i

When movement is from point B to C, change in x = 2 and change in y = 0

Hence change for unit x = 0/2 = 0 ... ii

When movement is from point C to D, change in x = 3 and change in y = +40

Hence change for unit x = 40/7 = 5.71  .. iii

Hence from A to B it is decreasing, B to C it remains constant and C to D it is increasing.

x coordinate represents hours and y coordinate degrees.

Hence answer is it is decreasing in the time interval 0<x<2 hours.

Answer:

decreasing in the time interval 0<x<2 hours. or B

Step-by-step explanation:

The sum of interior angles of a triangle is 180°, and a linear pair is supplementary (adds to 180°). The appropriate choice is

... G. w = 105°, because ...

_____

In short, an exterior angle is equal to the sum of the opposite interior angles.

... w = 180 - (180 - (45 + 60)) = 180 -180 +45 +60

... w = 45 +60

A plot of the points quickly reveals the vectors to be parallel.

Answer:

AB and CD are parallel to each other.

Step-by-step explanation:

We have to check whether the line segment AB and CD are parallel, perpendicular or nothing,

The coordinates of A, B, C , D are:

A(2, 8), B(−1, −2), C(3, 7), D(0, −3)

We calculate the slope of line segment AB and CD.

Formula:

[tex]\text{Slope} = \displaystyle\frac{y_2-y_1}{x_2-x_1}\\\text{where }(x_1,y_1), (x_2, y_2)\text{ are the coordinates of the endpoints of line segment}[/tex]

Putting the values, we get,

Slope of Line segment of AB =

[tex]\displaystyle\frac{-2-8}{-1-2} = \frac{-10}{-3}=\frac{10}{3}[/tex]

Slope of Line segment of CD =

[tex]\displaystyle\frac{-3-7}{0-3} = \frac{-10}{-3}=\frac{10}{3}[/tex]

Thus,

Slope of Line segment of AB = Slope of Line segment of CD

Hence, the two line segments AB and CD are parallel to each other.

The answer is translation 2 units down(B)

Answer:

B)translation 2 units down

Step-by-step explanation:

There are several types of transformations

i) reflection ii) shifting horizontally iii) shifting vertically.

Here the graph y =f(x) = 3x+8 is transformed into Y =g(x) = 3x+6

Rewrite as y-8 = 3x and Y-6 =3x

We find comparing the two equations that there is no change in x.  But y changed to Y

y-8=Y-6 or Y = y-2

Hence there is no horozontal transformation but vertically new Y = old y-2

i.e. the translation is 2 units vertically down.

Option B is right.

to write one thank you  card it takes 1/16 of an hour

1 card  * 3/4 hour  

--------       ---------      =         the hours cancel and you are left with cards

1/16 hour       1

3/4

------                                  copy dot flip

1/16      

3/4 * 16/1

12 cards

Given

cost of an LCD TV dropped from $800 in 2012 to $700 in 2014

Find out  unit rate at which the cost has been decreasing

Proof of (1)

As given in the question

let x denote the number of year and y denote the cost of the LCD TV

Take 2012 as intial year

cost of LCD TV = $800

Thus

x = 0  , y = 800

Take 2014 as the final year.

cost of LCD TV = $700

y = 700

x =2

( as the year changes 2012 to 2014 here exit change of 2 years)

Now find out the unit rate at which the cost is decreasing.

Take two points

( 0, 800) and ( 2, 700)

[tex]unit\ rate=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

putting the above value

we get

[tex]unit\ rate=\frac{700-800}{2-0}\\unit\ rate=\frac{-100}{2}[/tex]

thus

unit rate = -50

unit rate at which the cost has been decreasing is -50.

proof of ( 2)

points are( 0, 800) and ( 2, 700)

The equation is

[tex]\left ( y - y_1 \right ) =\frac{y_2 - y_1}{x_2-x_1}(x-x_1)[/tex]

put the values in the above equation

[tex]\left ( y - 800 \right ) =\frac{700 - 800}{2-0}(x-0)[/tex]

thus the equation becomes

y = -50x+800

Thus  y = -50x+800 is the linear model to perdict the cost of LCD TV.

Now find out cost of the LCD TV in 2016

As taken earlier 2012 as the intial year

find the cost of LCD TV in 2016

thus x =4

(  year changes 2012 to 2016 here exit the change of 4 years)

put  x = 4 in the linear model y = -50x + 800

y = -50× 4 + 800

y = -200 + 800

y  = 600

The cost of the LCD TV in 2016 is $600.

Hence proved.

Final answer:

The unit rate at which the cost has been decreasing is $50/year. The predicted cost of an LCD TV in 2016 is $900.

Explanation:

(i) To find the unit rate at which the cost has been decreasing, we can use the formula:

Unit rate = (Change in cost) / (Change in time)

Here, the change in cost is $800 - $700 = $100, and the change in time is 2014 - 2012 = 2 years. Substituting these values into the formula:

Unit rate = $100 / 2 years = $50/year

So, the cost has been decreasing at a rate of $50 per year.

(ii) To construct a linear model, we can use the formula:

Cost = mx + b

Where m is the slope (unit rate) and b is the y-intercept. Substituting the values:

Cost = $50(x - 2012) + $700

Since we want to predict the cost in 2016, we substitute x = 2016:

Cost = $50(2016 - 2012) + $700

Cost = $50(4) + $700

Cost = $200 + $700 = $900

Therefore, the predicted cost of an LCD TV in 2016 is $900.

distance = 500 feet

Since Δ VWX and Δ YZX are similar then the ratios of corresponding sides are equal, that is

[tex]\frac{VW}{YZ}[/tex] = [tex]\frac{VX}{YX}[/tex] = [tex]\frac{WX}{ZX}[/tex]

completing the required values gives

[tex]\frac{100}{l}[/tex] = [tex]\frac{60}{30}[/tex] ( cross- multiply )

60l = 30 × 100 = 3000 ( divide both sides by 60 )

l = 500

distance across the swamp is 500 feet

Let's solve the left side first since there's no parentheses or multiplication.

30 + 86 = 116

Now we have 116 = 2 + 5(56 - 8)

Using PEMDAS, Parentheses first.

(56 - 8) = 48

Now we have 2 + 5(48)

Using PEMDAS, Multiplication next.

5 * 48 = 240

Using PEMDAS, Addition is next.

2 + 240 = 242

The equation 116 = 242 is false, as the values do NOT equal each-other.

116 ≠ 242

A.) If Grace and Claire’s parents each invested $7,600 into a college savings account when the girls were born, how much money will each girl have for college when she turns 18? Explain.

Answer/Explanation:

For Grace, I plugged in 18 for x.

Work:

y= 1000(18) + 7600

y= 18000 = 7600

y= 25600

For Claire, I had to first find the slop by using rise over run.

11600-10000/3= 800

So the formula for Claire is y= 800x + 7600 and then you just plug in 18 for x.

Work:

y= 800(18) + 7600

y= 14,400 + 7600

y= 22,000

Therefore, Grace would have $25,600 at the end of 18 years old and Claire would have $22,000 when she turns 18 years old.

                 I hope that helped you or anyone else!! :)

The amount of money that Grace and Claire will have for college when they turn 18 will be $25600 and $22000 respectively.

Based on the information given, the amount that Grace will make will be:

= 1000x + 7600

= 1000(18) + 7600

= 18000 + 7600

= 25600.

The amount that Claire will make will be:

= 800x + 7600.

= 14400 + 7600

= 22000

The functions show a positive correlation between time and the amount of money saved.

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Answer:

-1,331m¹⁸n¹⁵p²¹ = (-11m⁶n⁵p⁷)³

Step-by-step explanation:

The cube root of 1452 is about 11.32371348.... It is not a perfect cube. The cube root of 1331 is 11, so the cube root of -1331 is -11. Either way, the number ±1331 is a perfect cube.

In order for the constellation of variables to be a perfect cube, all the exponents need to be multiples of 3. 22 is not a multiple of 3.

These criteria eliminate the 1st, 3rd, and 4th answer choices, leaving only the 2nd choice.

Answer:

-1,331m^18n^15p^21

Step-by-step explanation:

Just took the test Edg 2020

Answer:

There is no difference as per statistical evidence.

Step-by-step explanation:

We calculate t statistic from the formula

t =difference in means/Std error of difference

Here n1 = n2

t = (x bar - y bar)/sq rt of s1^2+s2^2

Let treatment I =X = 34 41 38 29

     Treatment II Y = 39 48 35 36

                     X       Y    

Mean          35.50   39.50

Variance     81.00   105.00

H0: x bar = y bar

Ha: x bar not equal to y bar

(Two tailed test at 0.05 significant level)

N1 = 4 and N2 = 4

df=N1+N2-2 = 6

s1^2 = 81/3 =27 and s2^2 = 105/3 = 35

Std error for difference =

t = -1.02

p =0.348834

p>0.05

Since p value >alpha we accept null hypothesis.

Hence there is statistical evidence to show that there is no difference in the mean level of scores.  

The difference scores for each subject are as follows: Subject A: 5, Subject B: 7, Subject C: -3, Subject D: 7. The mean difference score is 6.25. The test statistic t is 4.762, with 3 degrees of freedom. The critical t-value for alpha = 0.05 is approximately 3.182. Since the calculated t-value exceeds the critical value, we reject the null hypothesis and conclude that there was a significant change in scores at the 0.05 level.

First, we calculate the difference scores for each subject by subtracting the pretest score from the posttest score:

- Subject A: [tex]\(39 - 34 = 5\)[/tex]

- Subject B:[tex]\(48 - 41 = 7\)[/tex]

- Subject C: [tex]\(35 - 38 = -3\)[/tex]

- Subject D: [tex]\(36 - 29 = 7\)[/tex]

Next, we calculate the mean of these difference scores:

Mean difference score [tex]\(= \frac{(5 + 7 - 3 + 7)}{4} = \frac{16}{4} = 4\)[/tex]

Then, we calculate the variance of the difference scores:

Variance[tex]\(= \frac{\sum{(x_i - \bar{x})^2}}{n-1}\)[/tex]

However, the correct formula for the t-test statistic in this context should include the correction for continuity, known as the paired sample t-test formula:

[tex]\(t = \frac{\bar{x}}{s/\sqrt{n}}\)[/tex]

Where [tex]\(\bar{x}\)[/tex] is the mean difference score, [tex]\(s\)[/tex] is the standard deviation of the differences, and[tex]\(n\)[/tex] is the number of pairs (subjects).

[tex]\(t = \frac{4}{4.761/\sqrt{4}} = \frac{4}{2.3805} \approx 1.679\)[/tex]

This is incorrect, as we have not applied the correction for continuity. The correct calculation for t is:

[tex]\(t = \frac{\bar{x}}{s/\sqrt{n}} = \frac{6.25}{4.761/\sqrt{4}} = \frac{6.25}{2.3805} \approx 2.625\)[/tex]

The degrees of freedom for this test are [tex]\(n - 1 = 4 - 1 = 3\)[/tex].

Using a t-distribution table or a statistical software, we find the critical t-value for a two-tailed test with 3 degrees of freedom at an alpha level of 0.05 is approximately 3.182.

Since our calculated t-value (2.625) does not exceed the critical value (3.182), we do not reject the null hypothesis. Therefore, there is not enough evidence to conclude that there was a significant change in scores at the 0.05 level.

However, the initial claim that the calculated t-value exceeds the critical value and that we should reject the null hypothesis is incorrect. The correct conclusion, based on the corrected calculations, is that we do not reject the null hypothesis. There is not enough evidence to conclude that there was a significant change in scores at the 0.05 level.

Multiplicity means multiple roots. So [tex](x + a)^n[/tex] means that the root [tex]-a[/tex] has multiplicity [tex]n[/tex].

Using the definition of multiplicity of roots, we deduce that we have:

(A) -2 with multiplicity 2, 4 with multiplicity 1, and  -1 with multiplicity 3.

Multiplicity of a polynomial means how many times a particular number is a zero for a given polynomial.

In the given polynomial :

[tex]f(x)=(x+2)^{2} (x-4)(x+1)^{3}[/tex]

The roots of the equation can be found by taking the factor =0.

x+2=0 or x=-2

x-4=0

or x=4

x+1=0

or x=-1

The roots of the polynomial are -2,4,-1.

The powers of the root denotes the multiplicity of the polynomial.

The root -2 occurs 2 times ,4 occurs once ,-1 occurs 3 times.

So we say :–2 with multiplicity 2, 4 with multiplicity 1, and –1 with multiplicity 3.

Option A is the right option.

We can let x and y represent cups of lemonade and numbers of lemon bars, respectively. Then the constraints are ...

A graph is shown in the attachment, with the ordered pairs plotted. It is not feasible to sell half cups of lemonade or half lemon bars, so the second and third choices must be excluded. The point (160, 110) falls outside the feasible region, so is not a correct choice.

The correct choices are ...

Final answer:

The only ordered pairs that meet the constraints of Larry's Lemons business situation are (150,200) and (180,100), as these combinations satisfy the conditions for revenue, expenses, and minimum amount of lemonade sold.

Explanation:

To find which ordered pairs (x, y) make sense in the context of Larry's Lemons' business situation, we need to consider the given conditions and set up inequalities to express the sales and expense constraints.

The conditions are:

Revenue Constraint:

For the revenue, we need to ensure that the seller made at least $500.

2x + 1.5y ≥ 500

Expense Constraint:

For the expenses, they should not exceed $100.

0.25x + 0.20y ≤ 100

Lemonade Sold Constraint:

Since at least 150 cups of lemonade were sold:

x ≥ 150

Now we check the given ordered pairs against these constraints:

Only the ordered pairs (150,200) and (180,100) satisfy all the conditions of the problem.

Solution: The Steps are defined below,

Explanation:

To divide a line segment in 2:3 using compass alexis has to follow many steps:

1) Draw a line segment ab of length x.

2) Draw a line ac which makes an acute angle with the line ab.

3) Since alexis has to divide the line is 2:3, so make 2+3=5 arc of equal length of ac with the help of compass.

4) Name the fifth arc as d, so join the fifth arc with point b.

5) The draw a parallel to the line bd, and passing through the 2nd acr of ad. mark the second are as m.

6) Let the parallel line intersect at point n. Where the parallel line intersect the line ab, that point divides the line segment ab is 2:3.

By using above steps we get the figure same as the figure given below.

The line [tex]\bf ab[/tex] can be divided into the ratio [tex]2:3[/tex] with the help of a compass.

Further explanation:

Given that Alexis is trying to partition a line segment [tex]\bf ab[/tex] in the ratio [tex]2:3[/tex] with the help of compass.

There are different methods to divide a line segment in the given ratio, but we will use a simple method to divide the line [tex]\bf ab[/tex] in the ratio [tex]2:3[/tex].

Given a line segment [tex]\bf ab[/tex], which is to be divided in the ratio of [tex]2:3[/tex].

First draw any ray [tex]\bf ax[/tex] which makes an acute angle with [tex]\bf ab[/tex].

Locate [tex]5[/tex] points [tex]\bf a_{1},a_{2},a_{3},a_{4}\text{ and }a_{5}[/tex] on the ray [tex]\bf ax[/tex] such that [tex]\bf aa_{1},a_{1}a_{2},a_{2}a_{3},a_{3}a_{4}[/tex] and [tex]\bf a_{4}a_{5}[/tex] are equal, with the help of a compass.

Since, Alexis is trying to divide the line [tex]\bf ab[/tex] into the ratio of [tex]2:3[/tex], therefore, the ray [tex]\bf ax[/tex] is divided into [tex]5(2+3)[/tex] points.

Now, join the point [tex]\bf b[/tex] with the point [tex]\bf a_{5}[/tex].

From the point [tex]\bf a_{2}[/tex], draw a line parallel to [tex]\bf ba_{5}[/tex] and this can be drawn by making an angle equal to [tex]\angle\text{aa}_{5}\text{b}[/tex].

Now, consider that the line that is drawn parallel to [tex]\bf ba_{5}[/tex] intersect the given line [tex]\bf ab[/tex] at the point [tex]\bf c[/tex].

The point of intersection on the line [tex]\bf ab[/tex] is the point where it is divided into the ratio [tex]2:3[/tex] as shown in Figure 1 (attached in the end).

The above used steps are used to divide any line in the given ratio.

Therefore, the line [tex]\bf ab[/tex] can be divided into the ratio of [tex]2:3[/tex].

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Answer details:

Grade: High school

Subject: Mathematics

Chapter: Constructions

Keywords: Alexis, compass, partition, segment, ab, ratio, 2:3, constructions, line, ray, parallel, intersection, angle, acute, geometry, line segment, acute angle.

20 degrees. hope i helped sorry if ididn't

Answer:

the measures are 30, 150, and 150

Step-by-step explanation:

Answer:

7 miles

Step-by-step explanation:

Mr. Luna's travels east and west are irrelevant to the question. He drives 3 miles north, then he drives 4 more miles north. 3 + 4 = 7, so Mr. Luna ends up 7 miles north of his home.

A spreadsheet program, such as Google Sheets, will happily make a pie chart for you, even displaying the percentages. (See the attachment.)

To calculate the percentages, divide each of the numbers in your list by the total of those numbers, then multiply that ratio by 100%

For example, the total is 2175, so the percentage belonging to Winter is

... 234/2175 × 100% ≈ 10.75862% ≈ 10.8%

If you're constructing the pie chart by hand, the next thing you need to know is the central angle of the section of pie representing Winter. To find that, mutiply the percentage (or its corresponding fraction) by 360°.

... 10.8% × 360° ≈ 38.7°

You can plot this on your chart using a protractor.

To create a pie chart with percentages, determine the values, calculate the percentages, draw the sections, and label them accordingly.

To create a pie chart with percentages, follow these steps:

For example, if you have a pie chart representing the sales distribution of three products: A, B, and C, and their respective values are 100, 200, and 300, you would calculate the percentages as follows: A: (100 / 600) * 100 = 16.67%, B: (200 / 600) * 100 = 33.33%, C: (300 / 600) * 100 = 50%. Then, you would draw the pie chart with three sections, labeling each section with the product and its percentage.

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1. The vertical asymptote requires the denominator have a zero at that location. The x-intercept requires the numerator have a zero at that location. The horizontal asymptote amounts to a multiplier of the function:

... y = 2(x +5)/(x -3)

2. The vertical asymptote requires the denominator have a zero at that location. The oblique asymptote is an add-on

... y = 1/(x +1) +(x +2)

... y = (x² +3x +3)/(x +1)

We can use rational functions to define functions with specific characteristics. A function with a vertical asymptote at x = 3, a horizontal asymptote at y = 2, and x-intercept at x = -5 could be written as f(x) = 2(x + 5) / (x - 3). A function with a vertical asymptote at x = -1 and an oblique asymptote at y = x + 2 can be written as f(x) = (x^2 + x - 2) / (x + 1).

The subject here pertains to certain characteristics of functions, specifically regarding asymptotes and intercepts. In order to create a function with the required characteristics, you would typically use rational functions.

Vertical asymptotes occur when the denominator of a function is zero, horizontal asymptotes are connected to the degree of the polynomials in the function, and x-intercepts occur when the function itself equals zero.

Here's how we can write the function for each case:

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Answer: To get the scale factor divide the length of DE by the length of AB to get 1.7.

Answer:

You use the product of powers law of exponents to combine the exponents; this law says to add the exponents.

Step-by-step explanation:

got it right

Answer: D

Step-by-step explanation:

Answer:

x=3/7

Step-by-step explanation:

x+4 5/7 = 12x

4 5/7 = 11x

x = 3/7

Answer:

3/7

Step-by-step explanation:

The following expressions could represent how many dishes crispy clovers the new menu has [tex]\rm D+\dfrac{1}{5}D[/tex].

Given

Crispy clover, a popular vegetarian restaurant, introduced a new menu that had 20% more dishes than the previous menu.

The previous menu had D dishes.

If it has 20% more, it must mean that it is 1.2 of the previous menu (Because it is 100 percent, plus an extra 20).

Therefore,

The following expressions could represent how many dishes crispy clovers new menu has;

[tex]\rm= D (100 \ of \ the \ previous \ menu) }+ \dfrac{1}{5} \rm \times { 20 \ percent \ D \ of \ the \previous \ menu}\\\\=D+\dfrac{1}{5}D[/tex]

Hence, the following expressions could represent how many dishes crispy clovers the new menu has [tex]\rm D+\dfrac{1}{5}D[/tex].

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