Solve for x: 2^4x + 2^4+3 = 144

Both Tetsuya and Mason are interested in finding the probability that a red marble will be chosen out of the bag. We will decide if either of them is correct.

In statistics, what does a probability mean?

Tetsuya  P( R) = 4/17

Mason  P( R ) = 1 - 4/18

P ( R) = Favorable outcome/ possible outcome

The number of favorable outcomes is the number of red marbles in the bag, so it is equal to 4. The number of possible outcomes is the number of all marbles in the bag. Let's find it.

                     4 + 7 + 5 + 2 = 18

We have enough information to find the probability that a red marble will be chosen P(R).

                              P( R) = 4/18

Tetsuya used the wrong number of possible outcomes in the denominator. Let's look at Mason's solution. They wanted to use the probability of complementary events. However, they found the probability that a selected marble is not red.

                   Mason  P( R ) = 1 - 4/18

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ΔPRQ ≅ ΔTRS under both (side-side-side) SSS condition and (angle-side-angle) ASA condition.

So,

Given: ∠P ≅ ∠T, ∠S ≅ ∠Q, TR ≅ PR, RP ≅ RQ, RT ≅ RS, PQ ≅ TS

To Prove: Δ PRQ ≅ ΔTRS

So, ΔPRQ ≅ ΔTRS under both SSS condition and ASA condition.

Therefore, ΔPRQ ≅ ΔTRS under both SSS condition and ASA condition.

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Answer:

B.  -2  <  x  <  4

Step-by-step explanation:

The function is (x +2)/(x-4) and we are asked to determine the x interval when the inequality (x +2)/(x-4) < 0 is satisfied

1. Find the critical points

The critical points are

x = - 2 when the numerator becomes -2 + 2 = 0, and f(x) = 0

and

x = 4 when the denominator becomes 4 - 4 = 0 and (x + 2)/0 is undefined

So critical points are -2 and 4

2. Use these critical points to divide the number line into intervals
The intervals are -∞ < x < -2, -2 < x < 4 and 4 < x < ∞
these can be re-written as x < -2, -2 < x < -4 and x > 4

3. For each  of these intervals take a test point and see if it satisfies that inequality
(a) -∞ < x < -2    Take x = -3 which is less than -2
We get f(-3) =  (-3 + 2)/(-3 -4) = -1/-7 which is + 1/7 and does not satisfy the (x+2)/(x-4) < 0 inequality.

So we can discard this interval
(b) For interval -2 < x < 4, Let's take x = 0
(0+2)/(0-4) = 2/(-4) = -1/2 = -0.5 which does satisfy the inequality
(x+2)/(x-4) <0, so this is a valid interval

(c) For interval x > 4, take x = 5
(x+2)/(x-4 ) = (5 + 2)/(5-4) = 7/1 = 7 and does not satisfy (x+2)/(x-4 ) < 0

So the interval which satisfies the inequality is -2 < x < 4

                         Expression                               Property used

Step 1   1 x 3/2 x (-5) x 3 x (-4)        Inverse Property of Multiplication

Step 2  1 x (-4) x 3/2 x (-5) x 3        Associative Property of Multiplication

Step 3  1 x [ (-4) x 3/2 ] x (-5) x 3  Commutative Property of Multiplication

Step 4  1 x [-6] x (-5) x 3                Identity Property of Multiplication

Step 5  -6 x (-5) x 3

The inverse property of multiplication refers to multiplying a number with it's reciprocal. This is applied while transforming from division to multiplication as division is the inverse of multiplication.

Associative property of multiplication is applied while multiplying several values, Hence rearranging the terms will still give same answer.

This property has it that the order of the multiplication do not matter, hence multiplication can start with any order.

This property refers to multiplication of a value with one, such multiplication usually return same value as the product.

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Option D is the correct answer.

Triangle Inequality Theorem states that the sum of two side lengths of a triangle is always greater than the third side.

If this is true for all three combinations of three combinations of side lengths.

i.e., a + b > c , b + c > a and c + a > b then the length form a triangle.

(A) 10 + 11 = 21 > 20

And 11 + 20 =31 > 10

And, 20 + 10 = 30 > 11

So, it forms a triangle

(B) 14 + 16 = 30 >28

And 16 + 28 = 44 >14

And 28 + 14 = 42 > 16

So, it forms a triangle

(C) 35 + 45 = 80 > 75

And 45 + 75 = 120 > 35

And 75 + 35 = 110 > 45

So, it forms a triangle

(D) 41 + 55 = 96 < 98

It does not form a triangle

because sum of two sides lengths is less than the third side length.

Hence, Option D is the correct answer

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the function f(x) is invertible on the intervals:

B: -7 ≤ x ≤ -4

C : 1 ≤ x ≤ 3

A function f(x) can only be invertible if it is one-to-one in its domain, where one-to-one means that each output is related to only one input.

So, if for two different values x₁ and x₂ we have that:

f(x₁) = f(x₂) = y, then the function is not invertible.

So here what we need to do is identify in which interval is the function one-to-one.

If you look at the last option, we have the interval [1, 3]

We have that f(1) = 2, and then the function increases until f(3) = 4

In that interval the function is cleraly one-to-one, so, on that interval, we can find an inverse function to f(x).

Similarly, on the second option [-7, -4] we also can see that:

f(-7) =  2

And then the function increases until f(-4) = 4

Then the function is also invertible on that interval.

We conclude that the function f(x) is invertible on the intervals:

B: -7 ≤ x ≤ -4

C : 1 ≤ x ≤ 3

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Answer:-3<x<-1 is incorrect, all else are correct.

Step-by-step explanation:

The solution to the logarithmic equation is - 10 / 3.

Logarithms are examples of trascendent functions, that is, function that cannot be described in algebraic terms. Herein we must take advantage of logarithm properties to solve for x in the expression given. The complete procedure is shown below:

㏒₃ (3 · x - 2) - ㏒₃ (x + 2) = 2                        Given

㏒₃ [(3 · x - 2) / (x + 2)] = 2                             Logarithm of a division of functions

(3 · x - 2) / (x + 2) = 3²                                    Definitions of power and logarithm

(3 · x - 2) / (x + 2) = 9                                      Definition of power

3 · x - 2 = 9 · (x + 2)                                        Definition of division / Compatibility with multiplication / Existence of the multiplicative inverse / Associative and modulative properties

3 · x - 2 = 9 · x + 18                                         Distributive property

- 20 = 6 · x                                                       Compatibility with addition / Existence of the additive inverse / Associative, commutative and modulative properties

6 · x = - 20                                                       Symmetric property

x = - 20 / 6                                                       Definition of division / Compatibility of multiplication / Existence of the multiplicative inverse / Associative, commutative and modulative properties

x = - 10 / 3                                                        Simplification of fractions / Result

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The geometric mean between pair of numbers 8√2/3 and 4√2/3 is 2.67

We know that for a sequence of numbers x1, x2,.., xn

The formula of geometric mean is,

[tex]\Pi=\sqrt[n]{x_1.x_2.x_3...x_n}[/tex]

In this question, we have been given two numbers 8√2/3 and 4√2/3

We need to find the geometric mean between pair of numbers, 8√2/3 and 4√2/3.

Using the formula for the geometric mean,

[tex]\Pi=\sqrt[n]{x_1.x_2.x_3...x_n}[/tex]

In this question, we have two numbers.

So, the value of n is 2.

[tex]\Pi=\sqrt{\frac{8\sqrt{2} }{3} \times \frac{4\sqrt{2} }{3}} \\\\\Pi=\sqrt{\frac{32\times 2}{9} } \\\\\Pi=\sqrt{\frac{64}{9} } \\\\\Pi=\frac{8}{3}\\\\\Pi=2.67[/tex]

Therefore, the geometric mean between pair of numbers 8√2/3 and 4√2/3 is 2.67

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The distance between the point  [tex]\frac{4}{5}[/tex] and [tex]\frac{-1}{6}[/tex] is [tex]\frac{9}{10}[/tex] or 0.9.

Given point are [tex]\frac{4}{5}[/tex] and [tex]\frac{-1}{6}[/tex]

In decimal form,

[tex]\frac{4}{5}=0.8[/tex]

[tex]\frac{-1}{6} =-0.1[/tex]

Plot that two point on a number line.

Number line means a straight line with numbers arranged at equal segments or intervals throughout its length is known as a number line. A number line is often shown horizontally and can be extended indefinitely in any direction.

In the figure we can see the number plotting.

The points between those points is the distance between the  [tex]0.8[/tex] and [tex]-0.1[/tex].

Therefore, the distance between the point  [tex]\frac{4}{5}[/tex] and [tex]\frac{-1}{6}[/tex] is [tex]\frac{9}{10}[/tex] or 0.9.

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The direct variation equation between x and y is 4x = y.

Here, we are given that x and y are in direct variation. In other words, this means that x and y are directly proportional to each other.

Thus, if k is a constant term, then the relation between x and y can be represented as-

x = ky

Now, we are given that if y = 12, x = 3

⇒ 3 = 12k

12k = 3

k= 3/12

k= 1/4

Thus, we found the value of k, the relation will become-

x = y/4

or 4x = y

Thus, the direct variation equation between x and y is 4x = y.

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The interval using two different representations are (-∝, 11) ∪ (0, ∝) and {x | x < −11 or x > 0}

The interval is given as

x < −11 or x > 0

Inequalities like < and > are represented using the parenthesis ( and )

Also, the term "or" mean union and it is represented using ∪

So, one representation of the interval can be represented as

(-∝, 11) ∪ (0, ∝)

Another representation of the interval is {x | x < −11 or x > 0}

Hence, the interval using two different representations are (-∝, 11) ∪ (0, ∝) and {x | x < −11 or x > 0}

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Answer:

Use This it will help

Step-by-step explanation:

Based on the scale of the drawing, and the actual width, the width of a park in a drawing would be 14 inches.

A single inch of the width of a park in a drawing, is 5 yards of the width of the actual park.

This means that a single inch equates to 5 yards.

If there were therefore 70 yards in the actual park, the width of the drawing is:
= 70 / 5

= 14 inches

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Answer:

Hello,

Step-by-step explanation:

Do you remember the formula

[tex]\boxed{a^3-b^3=(a-b)(a^2+ab+b^2)}\\\\\\27x^6y^9-64=(3x^2y^3)^3-4^3\\\\=(3x^2y^3-4)((3x^2y^3)^2+(3x^2y^3*4)+4^2)\\\\\\=(3x^2y^3-4)(9x^4y^6+12x^2y^3+16)\\[/tex]

Answer:

I want to say False.

Step-by-step explanation:

I would think that the opposite of 4 would be -4.

Good luck! :)

Answer:

True

Step-by-step explanation:

4 = 4/1 1/4 is the reciprocal or the opposite of 4

Answer: Area, S = 36 [tex]cm^{2}[/tex]

Step-by-step explanation:

What is the Area of a Triangle?

The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e.

                               A = 1/2 × b × h.

Given data,

Area of shaded triangle S = 1/2 *b*h

We can write,

                    Area = S

                     Base = b

                     Height = h

we get,

                Base b = 6cm

                Height h = 12cm

h=12 cm  height of isosceles triangle ,

           Area = 1/2 * b*h

We can substitute b and h values,        

            Area= 1/2* 6 * 12

                    = 1/2 * 72

                    = 36

So, Area, S = 36 [tex]cm^{2}[/tex]

Therefore,

                          Area, S = 36 [tex]cm^{2}[/tex]

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Answer:

===============================================

Reason:

In the first graph, it is possible to draw a single vertical line through more than one point on the blue graph. For instance, draw a vertical line through 1 on the x axis.

Because this happens, we consider the graph to "fail the vertical line test". This leads to the graph not being a function.

A function is only possible if any input in the domain leads to exactly one and only one output.

The input x = 1 leads to multiple outputs y = 3 and y = -3 simultaneously.

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

On the other hand, the second graph doesn't have this issue happening. It's impossible to draw a single vertical line through more than one point on the blue graph.

Therefore, this graph passes the vertical line test and it is a function. Whatever x input you pick, it leads to one y output only.

We have proved that the West Texas Triangle formed is an isosceles triangle.

The approximate coordinates of three locations is given.

We have to prove that the West Texas Triangle is an isosceles triangle.

What is a triangle ?

A triangle is a 3-sided shape. There are three sides and three angles in every triangle, some of which may be the same.

As per the question ;

The West Texas Triangle is an isosceles triangle.

1st location has coordinates (31.9°N , 102.3°W)

2nd location has coordinates (32.7°N , 99.3°W)

3rd location has coordinates (31.4°N , 100.5°W)

Let's represent 1st location as A , 2nd location as B and 3rd as C.

Let's check it.

Using the distance formula ;

AB = √ [ (32.7 - 31.9) ² + (102.3 - 99.3)²]

AB = √ (0.8² + 3²)

AB = √9.64

AB = 3.10

Similarly ;

BC = √ (1.3² + 1.2²)

BC = √3.13

BC = 1.8

And

AC = √(0.5² + 1.8²)

AC = 1.8

The two sides are of equal length , so , we can conclude that this is an isosceles triangle.

Thus , the West Texas Triangle formed is an isosceles triangle.

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Answer:

x<-3

see the number line attached

make sure the number line has an open circle at -3

Hello,

[tex] - 9x + 9 > 36[/tex]

[tex] - 9x > 36 - 9[/tex]

[tex] - 9x > 27[/tex]

[tex]x < \frac{27}{ - 9 \:} \: \: because \: - 9 < 0[/tex]

[tex]x < - 3[/tex]

The local grocery store mixed 13 oz of planters mixed nuts with 19.5 oz of emerald mixed nuts to get 32.5 oz bags of mixed nuts that contain 56% peanuts.

An equation is an expression that shows the relationship between two or more numbers and variables.

Let x represent the amount of planters mixed nuts and y represent the amount of emerald mixed nuts, hence:

x + y = 32.5   (1)

Also:

0.5x + 0.6y = 0.56(32.5)       (2)

From both equations:

x = 13, y = 19.5

The local grocery store mixed 13 oz of planters mixed nuts with 19.5 oz of emerald mixed nuts to get 32.5 oz bags of mixed nuts that contain 56% peanuts.

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Answer: x = 5, 7

Step-by-step explanation:

       This equation is already factored for us, so we can split it into two separate equations set equal to 0 and solve for x.

Given:

(x - 7)(x - 5) = 0

Equation #1:

x - 7 = 0

x = 7

Equation #2:

x - 5 = 0

x = 5

Solution:

x = 5, 7

Answer:

x = 5, x = 7

Step-by-step explanation:

(x - 7)(x - 5) = 0 ← in standard factored form

equate each factor to zero and solve for x

x - 7 = 0 ⇒ x = 7

x - 5 = 0 ⇒ x = 5

Answer:

Q= ( -4 -9)

Step-by-step explanation:

Using the midpoint formula

X1 + X2 Y1 + Y1

----------, ---------

2. 2

The midpoint QR = (2, -5) and R = (-8, 1) Q =>(x, y)

= x + 8. y - 1

------------ = 2 and -------- = -5

2 2

=> (4 = x + 8) and( -10= y-1)

=> 4 -8 = x and -10+1= y

Q(x, y) = (-4, -9)

The value for x = -2,0,1 and 4 is -5,1,4,13 respectively.

To "solve for x" is to determine the value of x that would result in the equation being true. Think about the following equation: x plus 1 = 3. You would have to figure out a value for x such that it produces three when one is added in order to try to solve it.

For x = -2,

3x+1

= 3(-2)+1

= -6 +1

= -5

For x = 0

3x+1

= 3(0) + 1

= 1

For x = 1

3x+1

= 3(1) + 1

= 4

For x = 4

3x+1

= 3(4) + 1

= 13

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Answer:

Step-by-step explanation:

58 + 66(x-1) + 50 = 1428cents

66(x-1) + 108 = 1428

66(x-1) =1320

x-1 = 20

x = 21 mins

Answer:

561

Step-by-step explanation:

168/3=56 (10 percent of armandos pull ups)

56 x 10 = 560 (100 percent of armandos pull ups)

Now it is equal. We are looking for greater than armando. You would need 561 pull ups to have more

Answer: 1st option

Step-by-step explanation:

Black to Red is the transformation

C to C'

I to I'

Q to Q'

G to G'

1st, graph is moved up by 1 unit(+1 in x direction)

Then, graph is moved to right by 5 units(+5 in y direction)

(1, 5) is the change ==> 1st option

The 32th term of the arithmetic sequence AP is found to be -146.

An arithmetic progression (AP) is a succession in which each successive term has the same distinctions.

Now, to answer the question;

The given sequence of the numbers is-

9,4,-1,-6,-11, ...........

There are total 32 terms in the series for which we have to find 32nd number.

Let the first term be 'a₁' = 9.

Let the second term be 'a₂' = 4

Let the third term be 'a₃' = -1.

The common difference of between the two consecutive terms is equal of an AP is 'd'.

d = a₂ - a₁  

Substitute the values in the above equation;

d = 4 - 9

d = -5

Now, the 32th term is calculated by the nth term formula;

Now, substitute all the value in the formula,

a₃₂ = a + (n - 1) d

a₃₂ = 9 + (32 - 1)(-5)

a₃₂ = 9 - 160 + 5

a₃₂ = -146.

Therefore, the 32nd term of the arithmetic sequence is estimated as -146.

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