Factor the following expression so it is in simplest form.

6a – 4(2a + 3)

A.
–2(a – 6)


B.
–2(a + 6)


C.
2(7a – 6)


D.
2(7a + 6)

Answer:

[tex]3 {cos}^{2} x - {sin}^{2} x = 0[/tex]

[tex]3 - 3 {sin}^{2} x - {sin}^{2} x = 0[/tex]

[tex]3 - 4 {sin}^{2} x = 0[/tex]

[tex]4 {sin}^{2} x = 3[/tex]

[tex] {sin}^{2} x = \frac{3}{4} [/tex]

sin(x) = +(1/2)√3

x = π/3 + 2kπ or x = 2π/3 + 2kπ

x = -π/3 + 2kπ or x = -2π/3 + 2kπ

(k is an integer)

For a probability distribution to be represented, it is needed that P(X = 0) + P(X = 1) = 0.44. Hence one possible example is:

P(X = 0) = 0.40.

P(X = 1) = 0.04.

Discrete random variable to represent a probability distribution =

The sum of all the probabilities must be of 1, hence:

P(X = 0) + P(X = 1) + P(X = 3) + P(X = 4) + P(X = 5) = 1.

Then, considering the table:

P(X = 0) + P(X = 1) + 0.15 + 0.17 + 0.24 = 1

P(X = 0) + P(X = 1) + 0.56 = 1

P(X = 0) + P(X = 1) = 0.44.

Hence one possible value is:

P(X = 0) = 0.40.

P(X = 1) = 0.04.

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The  value of function f (α + β) is determined as  (3√3  +  1)/3,  (2√2 - 3)/3.

The value of f (α + β) is calculated by applying the following formula as follows;

The given functions;

for α:  x²  +  y² = 4

y is given as -1, the value of x is calculated as;

x²  +  (-1)² = 4

x²  +  1 = 4

x² = 3

x = √3

α = (√3, - 1)

for β:  x²  +  y² = 1

x is given as ¹/₃, the value of y is calculated as;

(¹/₃)² + y² = 1

¹/₉ + y² = 1

y² = 1 - ¹/₉

y² = ⁸/₉

y = √ (8/9)

y = 2√2/3

β = (¹/₃, 2√2/3)

The  value of function f (α + β) is calculated as;

f(α + β) =  (√3, - 1) + (¹/₃, 2√2/3)

f(α + β) = (3√3  +  1)/3,  (2√2 - 3)/3

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Answer:

5

Step-by-step explanation:

Just do as the problem says:

xy-y = (2)(5)-(5) = 10-5 = 5

Answer: 5

Step-by-step explanation: All you have to do is substitute x & y for their respective numbers. In this case, change x for 2 & y for 5, So xy-y becomes 2(5)-5, which equals 10-5, which is 5.

The equivalent sine and cosine values are as follows:

The relationship between sine and cosine angles is:

sinθ = cos(90 - θ)

This equation is known as the complementary angle identity for sine and cosine. It states that the sine of an angle is equal to the cosine of its complementary angle (90 degrees minus the original angle).

For example, if we have an angle theta = 30 degrees, its complementary angle would be 90 - 30 = 60 degrees.

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To find the length of arc LM, we need to use the formula for the circumference of a circle:

C = 2πr

In this case, the arc LM is a portion of the circumference of the circle. To find the length of the arc, we need to determine the angle at the center of the circle that corresponds to the arc.

Looking at the image you provided, we can see that angle LOM is a right angle (90 degrees). The angle at the center of the circle, LOCM, is half of the right angle, which is 45 degrees.

To calculate the length of arc LM, we need to find the fraction of the total circumference represented by the central angle LOCM. The fraction is given by:

Fraction of circumference = Angle at center / 360 degrees

Fraction of circumference = 45 degrees / 360 degrees

Fraction of circumference = 1/8

Now, we can calculate the length of arc LM:

Length of arc LM = Fraction of circumference * Circumference

Length of arc LM = (1/8) * (2πr)

Since the radius (r) is not given in the image, we cannot calculate the exact length of arc LM. However, we can provide the formula for the length of arc LM in terms of the radius:

Length of arc LM = (1/8) * (2πr) = πr/4

So, the length of arc LM is πr/4, where r is the radius of the circle.

If you have the value of the radius, you can substitute it into the formula to find the specific length of arc LM.

a. The surface area of the cylinder is 120π square units.

b. Assuming the diameter and the height of the cylinder are cut in half, the surface area of the cylinder would be reduced by a scale factor of 1/4.

In Mathematics and Geometry, the surface area (SA) of a cylinder can be calculated by using this mathematical equation (formula):

SA = 2πrh + 2πr²

Where:

Note: Radius = diameter/2 = 8/2 = 4 m.

By substituting the side lengths into the formula for the surface area (SA) of a cylinder, we have the following;

Surface area = 2πrh + 2πr²

Surface area = 2(π)(4)(11) + 2(π)(4²)

Surface area = 120π square units.

Part b.

Assuming the diameter and the height of the cylinder are cut in half, we have:

Surface area = 2(π)(4/2)(11/2) + 2(π)((4/2)²)

Surface area = 2(π)(2)(5.5) + 2(π)(2²)

Surface area = (π)(2)(11) + 2(π)(4)

Surface area = 22π + 8π

Surface area = 30π square units.

Scale factor = 30π/120π

Scale factor = 1/4.

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Note: "The diameter of the cylinder is 8 and the height is 11"

On performing , the "row-operation" 2R₁ + R₂ → R₂ on the matrix "M", the resulting new-matrix is [tex]\left[\begin{array}{ccc}5&1&-5\\10&0&-8\end{array}\right][/tex] .

The matrix "M" is given as : [tex]\left[\begin{array}{ccc}5&1&-5\\0&-2&2\end{array}\right][/tex],

We have to apply the Row-Operation : 2R₁ + R₂ → R₂, on the matrix "M",

Performing the row-operation "2R₁ + R₂ → R₂" on a matrix "M" means multiplying the first-row of "M" by 2, and then adding the resulting values to the second-row of "M" to produce a new value for each element in the second row.

This operation does not affect the first row of "M" and changes the values in the second row of "M".

⇒ [tex]\left[\begin{array}{ccc}5&1&-5\\2\times5+0&2\times 1-2&2\times(-5)+2\end{array}\right][/tex],

⇒ [tex]\left[\begin{array}{ccc}5&1&-5\\10+0&2-2&-10+2\end{array}\right][/tex],

⇒ [tex]\left[\begin{array}{ccc}5&1&-5\\10&0&-8\end{array}\right][/tex].

Therefore, the new-matrix is [tex]\left[\begin{array}{ccc}5&1&-5\\10&0&-8\end{array}\right][/tex].

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The sample size needed to be 99​% confident that the sample proportion will not differ from the true proportion by more than ​4% is given as follows:

n = 1037.

A confidence interval of proportions has the bounds given by the rule presented as follows:

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which the variables used to calculated these bounds are listed as follows:

The margin of error is given as follows:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

The confidence level is of 99%, hence the critical value z is the value of Z that has a p-value of [tex]\frac{1+0.99}{2} = 0.995[/tex], so the critical value is z = 2.575.

As we have no estimate, the parameter is given as follows:

[tex]\pi = 0.5[/tex]

Then the sample size for M = 0.04 is obtained as follows:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

[tex]0.04 = 2.575\sqrt{\frac{0.5(0.5)}{n}}[/tex]

[tex]0.04\sqrt{n} = 2.575 \times 0.5[/tex]

[tex]\sqrt{n} = \frac{2.575 \times 0.5}{0.04}[/tex]

[tex](\sqrt{n})^2 = \left(\frac{2.575 \times 0.5}{0.04}\right)^2[/tex]

n = 1037.

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Answer:

8% of $15,000 = .08 × $15,000 = $1,200

The total surface area of the cone is 427.04 square inches.

For a cone of radius R and slant height L, the total surface area is given by the formula:

S = π*R*L + π*R²

Where π = 3.14

In this case, we know that the radius of the cone is 8 inches, and the slant height of the cone is 9 inches.

Then the total surface area of the cone is:

S = 3.14*8in*9in + 3.14*(8in)²

S = 427.04 in²

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Therefore, Jan has taken the wrong slant height, he has taken height, h is a place of slant height, l. That's why he has got the wrong answer.

Given:,

r= d/2

22/2= 11m

Using the Pythagorean theorem, we can find the length:

l =[tex]11\sqrt{5}\\ = 24.6m[/tex]

Therefore, to find the surface area:

SA= 3.14(270.56 + 121)

=[tex]1229.5m^2[/tex]

Therefore, to find the volume:

V= [tex]1/3 \pi.r^2h[/tex]

V= [tex]2786.23m^3[/tex]

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The Complete Question

Jan INCORRECTLY finds the surface area of the cone using the following work. Explain Jan's error and find the correct volume AND surface area of the cone. 22 m SA = = url + ar?

You are approximately 3.4 miles from your starting point.

We  have,

To solve this problem, we need to use the Pythagorean theorem, which states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the length of the hypotenuse.

 |\

8 | \

 |  \

 |___\

    3

We can see that the three blocks north and 10 blocks south cancel each other out, so we only need to consider the 8 blocks west.

Using the distance formula, we can find that the distance traveled west is:

8 blocks × 2/10 mile per block = 1.6 miles

Now we can use the Pythagorean theorem to find the distance from the starting point:

distance = √(3² + 1.6²)

distance ≈ 3.4 miles

Therefore,

You are approximately 3.4 miles from your starting point.

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The requried, Multiplies(with switching factors.) area given below,

a.

693 x 83 = 57489

83 x 693 = 57489

The answer is 57489.

b.

910 x 45 = 40950

45 x 910 = 40950

The answer is 40950.

c.

38 x 84 = 3192

84 x 38 = 3192

The answer is 3192.

d.

409 x 89 = 36401

89 x 409 = 36401

The answer is 36401.

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Answer:

444523.45 square miles

Step-by-step explanation:

By using Heron's formula, we can easily find the area of the triangle formed by Miami, Bermuda, and San Juan, we need to use the lengths of the three sides of the triangle.

Let,

Side a: Distance from Miami to Bermuda = 1042 miles

Side b: Distance from Bermuda to San Juan = 965 miles

Side c: Distance from San Juan to Miami = 1038 miles

Now we can use Heron's formula to find the area of the triangle:

s =[tex]\frac{a+b+c}{2}[/tex]

s = [tex]\frac{1042 + 965 + 1038}{2}=1522.5[/tex] miles

A = [tex]\sqrt{s(s-a)(s-b)(s-c)}[/tex]

A = [tex]\sqrt{1522.5(1522.5-1042)(1522.5-965)(1522.5-1038)}=444523.45[/tex]

Therefore, the area of the triangle formed by Miami, Bermuda, and San Juan is approximately 444523.45 square miles.

Answer:

444,523.45 square miles (2 d.p.)

Step-by-step explanation:

To find the area of a triangle formed by the locations of Miami, Bermuda, and San Juan, use Heron's formula.

[tex]\boxed{\begin{minipage}{8 cm}\underline{Heron's Formula}\\\\$A=\sqrt{s(s-a)(s-b)(s-c)}$\\\\where:\\ \phantom{ww}$\bullet$ $A$ is the area of the triangle. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the side lengths of the triangle. \\ \phantom{ww}$\bullet$ $s$ is half the perimeter.\\\end{minipage}}[/tex]

Label the three sides of the triangle as 'a', 'b', and 'c', where 'a' is the distance from Miami to Bermuda (1042 miles), 'b' is the distance from Bermuda to San Juan (965 miles), and 'c' is the distance from San Juan to Miami (1038 miles):

To find the half perimeter, s, half the sum of the three side lengths:

[tex]\implies s=\dfrac{a+b+c}{2}=\dfrac{1042+965+1038}{2}=1522.5[/tex]

Substitute the values of a, b, c and s into Heron's formula and solve for area, A:

[tex]\begin{aligned}A&=\sqrt{s(s-a)(s-b)(s-c)}\\&=\sqrt{1522.5(1522.5-1042)(1522.5-965)(1522.5-1038)}\\&=\sqrt{1522.5(480.5)(557.5)(484.5)}\\&=444523.4468348...\\&=444523.45\; \sf miles^2\;(2\;d.p.)\end{aligned}[/tex]

Therefore, the area of the triangle formed by the three locations is 444,523.45 square miles, to two decimal places.

The slopes of the lines connecting all the points are distinct. Hence, they form a triangle.

To show that the points (2, 5), (5, 2), and (6, 6) form a triangle, we can calculate the slopes of the lines connecting these points. If the slopes are all distinct, then the points form a triangle.

Let's calculate the slopes:

The slope between (2, 5) and (5, 2):

m₁ = (y₂ - y₁) / (x₂ - x₁)

= (2 - 5) / (5 - 2)

= -3/3

= -1

The slope between (2, 5) and (6, 6):

m₂ = (y₂ - y₁) / (x₂ - x₁)

= (6 - 5) / (6 - 2)

= 1/4

The slope between (5, 2) and (6, 6):

m₃ = (y₂ - y₁) / (x₂ - x₁)

= (6 - 2) / (6 - 5)

= 4/1

= 4

Since the slopes of the lines connecting these points are all distinct (-1, 1/4, and 4), the points (2, 5), (5, 2), and (6, 6) form a triangle.

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The surface area of Grace's shipping box is 664 square inches.

We have,

The box can be divided into six rectangles, each of which represents the face of the box.

To find the surface area of the box, we need to find the area of each face and then add them all up.

The top and bottom of the box are both 10 inches by 14 inches, so the area of each of these faces is:

= 10 x 14

= 140 square inches

The sides of the box are both 14 inches by 8 inches, so the area of each of these faces is:

= 14 x 8

= 112 square inches

The front and back of the box are both 8 inches by 10 inches, so the area of each of these faces is:

8 x 10

= 80 square inches

To find the surface area of the whole box, we add up the areas of all six faces:

= 2(140) + 2(112) + 2(80)

= 280 + 224 + 160

= 664 square inches

Therefore,

The surface area of Grace's shipping box is 664 square inches.

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There are 23 boys and 20 girls in the class who cannot read a letter or a three-letter word.

The Venn diagram is drawn as shown in the attached image, and it shows that the number of boys and girls who cannot read a letter or a three-letter word is marked as (c+d) which represents 13 boys and 8 girls.

Given:

The class has 42 students.

12 boys and 15 girls are said to know how to read three-letter words in English.

19 boys and 17 girls are said to know how to identify all letters of the alphabet.

None of the students in the class is able to write a sentence of five words if dictated to them.

There are in total 22 boys and 20 girls in the class.

Read three-letter words    |

     |____________|______________|  

     |            |              |

     |  Boys (12) | Girls (15)   |   Can read three-letter words

     |____________|______________|

     |            |              |

     |  Boys (10) | Girls (13)   |   Cannot read three-letter words

_______|____________|______________|

     |            |              |

     |  Boys (9)  | Girls (13)   |   Can identify all letters

     |____________|______________|

     |            |              |

     |  Boys (13) | Girls (7)    |   Cannot identify all letters

     |____________|______________|

To find the number of boys and girls who cannot read a letter or a three-letter word, we need to add up the numbers in the two bottom circles:

Boys who cannot read a letter or a three-letter word: 10 + 13 = 23

Girls who cannot read a letter or a three-letter word: 13 + 7 = 20

Therefore, there are 23 boys and 20 girls in the class who cannot read a letter or a three-letter word.

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The number of 1/3 inch cubes required to fill the box is 62.

Volume = length × width × height

Height = 7/3

Length = 10/3

width = 8/3

Hence ,

Volume = 7/3 × 10/3 × 8/3 = 560/27

Size of cube = 1/3

Number of cubes required = Volume of box / size of cube

Number of cubes = 560/27 × 3/1

Number of cubes = 1680/27 = 62.22

Therefore, it will take 62 1/3 cubes to fill the box

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Answer: 0

Step-by-step explanation:

Put 0 in the front box.

this is because you cant divide 11 apples equally into 6 people; 11 isn't a multiple of 6

The property used to calculate x in (x + 2)(x + 6) = 0 is (a) the zero product property

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

(x + 2)(x + 6) = 0

The equation when expanded becomes

x + 2 = 0 or x + 6 = 0

In algebra, the zero product property states that

if ab = 0, then a = 0 or b = 0

using the above as a guide, we have the following:

The property used to calculate x is (a) the zero product property

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Answer:

[tex]\sf 2\sqrt{17}[/tex]

Step-by-step explanation:

To find the distance of a line using distance formula:

The line from point X intersect the line p at (0 , -3).

          ( -2 , 5) ⇒  x₁  = -2 & y₁ = 5

          (0 , -3)  ⇒ x₂ = 0    & y₂ = -3

    [tex]\boxed{\bf Distance = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}}[/tex]

                       [tex]\sf = \sqrt{-2-0)^2+(5-[-3])^2}\\\\=\sqrt{(-2-0)^2 + (5+3)^2}\\\\=\sqrt{(-2)^2+(8)^2}\\\\=\sqrt{4+64}\\\\=\sqrt{68}\\\\=\sqrt{2*2*17}\\\\=2\sqrt{17}[/tex]

The correct step to prove that [tex]BC^2 = AB^2 + AC^2[/tex] is:

By the cross product property, [tex]AC^2 = BC \cdot AD[/tex].

To prove that [tex]BC^2 = AB^2 + AC^2[/tex], we can use the triangle similarity and the Pythagorean theorem. Here's a step-by-step explanation:

Given triangle ABC with right angle at A and segment AD perpendicular to segment BC.

By triangle similarity, triangle ABD is similar to triangle ABC. This is because angle A is common, and angle BDA is a right angle (as AD is perpendicular to BC).

Using the proportionality of similar triangles, we can write the following ratio:

[tex]$\frac{AB}{BC} = \frac{AD}{AB}$[/tex]

Cross-multiplying, we get:

[tex]$AB^2 = BC \cdot AD$[/tex]

Similarly, using triangle similarity, triangle ACD is also similar to triangle ABC. This gives us:

[tex]$\frac{AC}{BC} = \frac{AD}{AC}$[/tex]

Cross-multiplying, we have:

[tex]$AC^2 = BC \cdot AD$[/tex]

Now, we can substitute the derived expressions into the original equation:

[tex]$BC^2 = AB^2 + AC^2$\\$BC^2 = (BC \cdot AD) + (BC \cdot AD)$\\$BC^2 = 2 \cdot BC \cdot AD$[/tex]

It was made possible by cross-product property.

Therefore, the correct step to prove that [tex]BC^2 = AB^2 + AC^2[/tex] is:

By the cross product property, [tex]AC^2 = BC \cdot AD[/tex].

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Answer:

A

Step-by-step explanation:

i do this all the time math is my thing

A total of 8 workers are required for a building time of 15 days.

In this problem we find a case of inverse relationship, in which the number of workers (n) is inversely proportional to building time (t), in hours. The situation is represented by following formulas:

n ∝ 1 / t

n = k / t

Where k is the proportionality ratio, which can be eliminated by building the following formula:

n₁ · t₁ = n₂ · t₂

If we know that n₁ = 10, t₁ = 12 and t₂ = 15, then the required number of workers is:

n₂ = n₁ · (t₁ / t₂)

n₂ = 10 · (12 / 15)

n₂ = 10 · (4 / 5)

n₂ = 8

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The resulting mixture contains approximately 64.5% alcohol, not 58.6% as you mentioned.

We can calculate the weighted average of the alcohol percentages based on the volumes of each component.

Let's figure out how much alcohol there is overall in the mixture:

Total alcohol is calculated as follows: (Volume of 45% alcohol) * (Property of 45% alcohol) + (Volume of 82% alcohol) * (Property of 82% alcohol) + (Volume of 92% alcohol) * (Property of 92% alcohol).

Total alcohol = (5 liters) * (45%) + (4 liters) * (82%) + (1 liter) * (92%)

Total alcohol = 2.25 liters + 3.28 liters + 0.92 liters

Total alcohol = 6.45 liters

Let's now determine the amount of alcohol included in the final mixture:

Alcohol content is calculated as (total alcohol / total mixture volume) * 100.

The mixture's total volume is  = 5 liters + 4 liters + 1 liter = 10 liters

Alcohol content: (6.45 liters / 10 liters) x 100 = 64.5%

Therefore, the resulting mixture contains approximately 64.5% alcohol, not 58.6% as you mentioned.

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Answer:

  (b)  $42.79

Step-by-step explanation:

You want the monthly interest charge on a balance of $1834.50 when the annual rate is 27.99%.

The monthly interest rate is the annual rate divided by 12. This means the interest charge is ...

  I = Prt . . . . . interest on P at annual rate r for t years

  I = $1834.50 × 0.2799 × 1/12 ≈ $42.79

Her monthly interest charge will be $42.79.

Answer:

13 glue sticks cost $25.22.

Step-by-step explanation:

4 blue sticks cost $7.76, this means that one glue stick costs

$7.76/4 = $1.94.

Let  be the cost of the glue sticks, and  be the number of glue sticks; then

We can use this equation to find the cost of 13 glue sticks; we just put  into our equation and it gives:

So 13 glue sticks cost $25.22.

62. The slope of the line is 1/8. the equation of line is y = 1/8x - 3/8

63. It has an infinite slope.  The equation of line is x = 3

64. The slope of the line is -5. The slope of the new line is 1/5. The equation of line is  y = 1/5x - 1

65. The slope of the line is 2/9. The slope of the new line is -9/2. The equation of line is y = -9/2x - 4

62) through: (3,0), parallel to y = 1/8x - 3. If a line is parallel to another, they have the same slope. Therefore the slope is 1/8.

For line equation⇒ (y - y1 = m(x - x1))

Substituting (3,0) and slope 1/8 gives us:

(y = mx + b) ⇒ y - 0 = 1/8 × (x - 3)

First, substitute the point and the slope into the equation: 0 = 1/8×3 + b

Then solve for b, the y-intercept: b = -3/8

⇒ y = 1/8x - 3/8

63) through: (3,-1), parallel to x = 0. A line parallel to x = 0 is a vertical line. A vertical line has the equation x = a, where a is the x-coordinate of any point on the line. A line through (3, -1) and parallel to x = 0 would simply be x = 3 or y = -1

64) through: (5, 0), perp. to y= -5x - 4. If a line is perpendicular to another, their slopes are negative reciprocals. The slope of y = -5x - 4 is -5, so the slope of the line we are looking for is -1/(-5) = 1/5

65) through: (-2, 5), perp. to y=2/9x - 1.  perpendicular  line to y = 2/9x - 1 has the slope that is the negative reciprocal of 2/9, which is -9/2. y = -9/2x - 4.

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