Lesson 7 Extra Practice Equivalent Expressions Simplify each expression. 1. 5x + 3x 8x 2. 2x + x 3x 3. 6x + 8x 14x 4. 9x + 3x 12x 5. 7x + 4x 11x 6. 10x + x 11x 7. 3x + 2 + 4x 7x + 2 8. 15x + 3 + x 16x + 3 9. 2x + 5 + 6x 8x + 5 10. 7x + 2x + 4 9x + 42 11. 18x + 3x + 9 21x + 9 12. 6x + x + 8 7x + 8 Factor each expression. 13. 21x + 15y 3(7x + 5y) 14. 2x + 14y 2(x + 7y) 15. 15x + 10y 5(3x + 2y) 16. 12x + 18y 6(2x + 3y) 17. 10x + 30y 10(x + 3y) 18. 6x + 14y 2(3x + 7y) 19. 8x + 16y 8(x + 2y) 20. 45x + 27y 9(5x + 3y) 21. 54x + 48y

Answer:

The scientific notation is a method of writing very large or very small numbers in a more compact and easy to read format. It is expressed in the form of a x 10^n, where "a" is a number between 1 and 10, and "n" is an integer.

To write 464000 in correct scientific notation, we need to move the decimal point to the left until we get a number between 1 and 10. We count the number of places we moved the decimal point, and that will be the exponent of 10.

So, we can write 464000 as:

4.64 x 10^5

Therefore, the correct scientific notation of 464000 is 4.64 x 10^5.

Step-by-step explanation:

Given:

Diagram is given.

Pair of opposite rays are:

Answers:

A, B, D, E

Nearly everything except choice C is true.

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

Explanation:

The value of a is 5, b is 4, c is 0, d is 3, e is 6 and R is 6 after following the long division method.

According to the question,

We have to divide 430 by 8 by following the long division method.

Note that when we follow long division method then the number by which we are dividing (divisor) is to be multiplied in such a way that the result remains less than the number in the dividend.

Now, we will solve it step by step.

In dividing 43 by 8, we have to multiply 8 by 5. Then, we get 40.

So, we have a = 5, b = 4 and c = 0.

Now, after this we have 30 and the number that we get after multiplication is 24. So, 3 has to be multiplied in 8 to get 24.

So, we have d = 3.

Now, when we will subtract 24 from 30, we will get 6.

So, we have e = 6.

And e is also the remainder, R. So, we have R = 6.

Hence, the values of a, b, c, d, e, and R are 5, 4, 0, 3, 6, and 6 respectively.

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Using trigonometric ratio it is proved that sin(B + 2C) + sin(C + 2A) + sin(A + 2B) = 4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2 when A + B + C = 180.

What is trigonometric ratio?

Triangle side length ratios are known as trigonometric ratios. In trigonometry, these ratios show how the ratio of a right triangle's sides to each angle. Sine, cosine, and tangent ratios are the three fundamental trigonometric ratios.

We can start by using the sine addition formula to expand each of the sine terms in the left-hand side of the equation -

sin(B + 2C) = sin(B)cos(2C) + cos(B)sin(2C) = 2sin(B)cos(C)²

sin(C + 2A) = sin(C)cos(2A) + cos(C)sin(2A) = 2sin(C)cos(A)²

sin(A + 2B) = sin(A)cos(2B) + cos(A)sin(2B) = 2sin(A)cos(B)²

Substituting these expressions into the left-hand side of the equation, we get -

2sin(B)cos(C)² + 2sin(C)cos(A)² + 2sin(A)cos(B)²

Factoring out the 2, we can rewrite this as -

2(sin(B)cos(C)² + sin(C)cos(A)² + sin(A)cos(B)²)

Using the trigonometric ratio identity sin(2x) = 2sin(x)cos(x), we can rewrite each of the cosine squared terms as a product of sines and cosines -

cos(C)² = (1/2)(1 + cos(2C)) = (1/2)(1 + 2cos(C)sin(C))

cos(A)² = (1/2)(1 + cos(2A)) = (1/2)(1 + 2cos(A)sin(A))

cos(B)² = (1/2)(1 + cos(2B)) = (1/2)(1 + 2cos(B)sin(B))

Substituting these expressions into the previous equation, we get -

2(sin(B)(1/2)(1 + 2cos(C)sin(C)) + sin(C)(1/2)(1 + 2cos(A)sin(A)) + sin(A)(1/2)(1 + 2cos(B)sin(B)))

Simplifying and grouping the terms, we get -

sin(B)sin(C)cos(C) + sin(C)sin(A)cos(A) + sin(A)sin(B)cos(B)

Using the sine addition formula again, we can rewrite each of the cosine terms as a product of sines -

cos(C) = sin(A + B)

cos(A) = sin(B + C)

cos(B) = sin(C + A)

Substituting these expressions into the previous equation, we get -

sin(B)sin(C)sin(A + B) + sin(C)sin(A)sin(B + C) + sin(A)sin(B)sin(C + A)

We can rearrange this expression by factoring out a sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 term -

sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 (cos(A) - cos(B) + cos(B) - cos(C) + cos(C) - cos(A))

Simplifying the terms in parentheses, we get -

sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 (0)

Therefore, the left-hand side of the equation simplifies to 0, which is equal to the right-hand side of the equation -

4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2

Therefore, we have proven that sin(B + 2C) + sin(C + 2A) + sin(A + 2B) = 4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2.

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Answer:

5x+7 > 17 matches with 3.

1+3x > -5 matches with 2.

2x+1 < 3 matches with 1.

5-3x > 11 matches with 4.

Step-by-step explanation:

The person is approximately 8.43 miles from home, and they need to walk in the direction of approximately 26.6 degrees east of north to head directly home.

To determine the distance from home, we can use the concept of vector addition. The person walks 4 miles west, which can be represented as a vector (-4, 0) in a coordinate plane. Then, they walk 5 miles southwest, which can be represented as a vector (-3.54, -3.54) since southwest is a combination of west and south. Adding these two vectors together gives us (-7.54, -3.54).

To find the magnitude or distance from home, we use the Pythagorean theorem. The distance is given by sqrt((-7.54)^2 + (-3.54)^2) = 8.43 miles (approximately).

To determine the direction to head directly home, we need to find the angle between the vector (-7.54, -3.54) and the positive x-axis. We can use trigonometry to find this angle. The tangent of the angle is given by the ratio of the y-coordinate to the x-coordinate, which is -3.54 / -7.54. Taking the inverse tangent of this value gives us approximately 26.6 degrees.

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The size of the largest interior angle of the field is 69.86°

The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.

Given that, a triangular field has boundaries of lengths 170m, 195m and 210m, we need to find the size of the largest interior angle of the field,

We know that, angle opposite to the largest side is largest,

Therefore,

The largest angle is opposite is 210 m side, (say c)

Using the cosine rule,

210² = 170²+195²-2(170)(195)cosC

CosC = 170²+195²-210² / 2(170)(195)

CosC = 22825 / 66300

CosC = 0.344268477

C = 69.86°

Hence, the size of the largest interior angle of the field is 69.86°

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The values of functions are

(a) f(0) = -3.                           (b) f(4) = 77.

(c) f(-4) = 45.                         (d) -f(x) = -4x² - 4x + 3.

(e) f(x+1) =  4x² + 12x + 1.       (f) f(-x) =  4x² - 4x - 3.

(g) f(3x) =  36x² + 12x - 3.

(h) f(x+h) =  4x² + 8xh + 4h² + 4x + 4h - 3.

The concept used in this question is evaluating a polynomial function for different values of the input variable. To do this, we substitute the given value into the function in place of the input variable and simplify the resulting expression.

Here we have

The function f(x) = 4x² + 4x-3.

To find the values of functions can be done by replacing x

(a) f(0) = 4(0)² + 4(0) - 3 = 0 + 0 - 3 = -3

(b) f(4) = 4(4)² + 4(4) - 3 = 64 + 16 - 3 = 77

(c) f(-4) = 4(-4)² + 4(-4) - 3 = 64 - 16 - 3 = 45

(d) -f(x) = -[4x² + 4x - 3] = -4x² - 4x + 3.

(e) f(x+1) = 4(x+1)² + 4(x+1) - 3 = 4(x² + 2x + 1) + 4x + 4 - 3 = 4x² + 12x + 1.

(f) f(-x) = 4(-x)² + 4(-x) - 3 = 4x² - 4x - 3.

(g) f(3x) = 4(3x)² + 4(3x) - 3 = 36x² + 12x - 3.

(h) f(x+h) = 4(x+h)² + 4(x+h) - 3 = 4(x² + 2xh + h²) + 4x + 4h - 3

= 4x² + 8xh + 4h² + 4x + 4h - 3.

Hence,

The values of functions are

(a) f(0) = -3.                           (b) f(4) = 77.

(c) f(-4) = 45.                         (d) -f(x) = -4x² - 4x + 3.

(e) f(x+1) =  4x² + 12x + 1.       (f) f(-x) =  4x² - 4x - 3.

(g) f(3x) =  36x² + 12x - 3.

(h) f(x+h) =  4x² + 8xh + 4h² + 4x + 4h - 3.

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Answer:

4, 8, 16,64 and 4096.

Step-by-step explanation:

We are already given: \(4096=2^{12}\)

To determine other whole numbers that can be raised to a power to obtain 4096, we apply the product rule of indices.

Product Rule of Indices: \(a^{xy}=(a^x)^y\)

Now 12 can be factored in the following ways where one of the terms must be a perfect square:

\(2^{12}=(2^2)^6=4^6\\\\2^{12}=(2^6)^2=64^2\\\\2^{12}=(2^4)^3=16^3\\\\2^{12}=(2^3)^4=(8^2)^2=8^{2*2}=8^4\\\\2^{12}=(2^{12})^1=4096^1 $(This is the trivial case)\)

Therefore, the other whole numbers that can be raised tp a power to obtain 4096 are: 4, 8, 16, 64 and 4096.

\(\huge\text{Hey there!}\)

\(\large\text{When you hear/see the word quotient in mathematics, think}\\\large\text{of its telling you to DIVIDE}\)

\(\large\text{So, that being said, your answer is: }\)

\(\boxed{\boxed{\mathsf{\bf -9 \div y\ \ or\ \dfrac{-9}{y}}}}\huge\checkmark\)

\(\text{Good luck on your assignment and enjoy your day!}\)

~\(\frak{Amphitrite1040:)}\)

The amount she invested in each account is:

$6,000 invested at 11%

$6,000 invested at 6%

Interest is the price you pay to borrow money or the cost you charge to lend money. Interest is most often reflected as an annual percentage of the amount of a loan. This percentage is known as the interest rate on the loan.

Here, we have,

formula: I = Prt.

here, we have,

Let x represent the amount invested at 11%

Hence:

.11x+.06(8000-x)=780

.05x+480=780

Collect like terms

.05x=300

Divide both side by .05x

x=300/.05

x=$6,000

In conclusion the amount she invested in each account is: $6,000 invested at 11% and $6,000 invested at 6%.

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To find the volume, we will use the formula;

V = πr² h

where v is the volume, r is the radius and h is the height

From the question, r = 9 h = 42 and π = 3.14

substitute the values into the equation

V = 3.14 x 9² x 42

= 3.14 x 81 x 42

=10682.28

V≈ 10682 ft³ to the nearest whole number

she could have scored the 10 points in 302400 ways.

The given parameters are

n= total points =10

r1 =one-point free throw = 1

r2 = two-point field goals = 2

r3 =  three-point field goals = 3

The number of ways (k) she could have scored the points is:

k=(n!)/(r1!×r2!×r3!)

The factorial function is a mathematical formula represented by an exclamation mark "!".

k= 10!/(1!×2!×3!)

k= 3628800/(1×2×6)

k= 302400

so.. she could have scored the 10 points in 302400 ways.

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Answer:

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

Steps should be self-explanatory. Happy to explain if anything is not clear.

Answer:

Step-by-step explanation:

1) 4v + 7 = 7, for v?

→ 4v + 7 = 7

→ 4v = 7 - 7

→ v = 0/4

→ [ v = 0 ]

2) 3(u + 4) = 27, for u?

→ 3(u + 4) = 27

→ 3u + 12 = 27

→ 3u = 27 - 12

→ u = 15/3

→ [ u = 5 ]

3) -20 = 10(h + 5), for h?

→ -20 = 10(h + 5)

→ 10h + 50 = -20

→ 10h = -20 - 50

→ h = -70/10

→ [ h = -7 ]

4) -10 = (n - 52)/5, for n?

→ -10 = (n - 52)/5

→ n - 52 = -10 × 5

→ n = -50 + 52

→ [ n = 2 ]

5) (48/x) - 3 = 3, for x?

→ (48/x) - 3 = 3

→ 48/x = 3 + 3

→ 6x = 48

→ x = 48/6

→ [ x = 8 ]

6) 8r = 2r + 42, for r?

→ 8r = 2r + 42

→ 8r - 2r = 42

→ r = 42/6

→ [ r = 7 ]

These are required values.

One of the function that satisfies all four statements is f(x) = 2 - \(e^{3-x}\) , for x ≤ 3.

The function that satisfies all four statements is a decreasing function with a horizontal asymptote y = 0 that intersects the x-axis at x = 3. One possible function that satisfies all four statements is:

f(x) = 2 - \(e^{3-x}\), for x ≤ 3

Explanation:

Statement 1, The domain of the function is x ≤ 3 because the expression \(e^{3-x}\) becomes undefined for x > 3.

Statement 2, The range of the function is y ≥ 0 because the exponential term \(e^{3-x}\) is always positive and subtracting it from 2 will always result in a non-negative value.

Statement 3, The x-intercept of the function is 3 because f(3) = 0, which means the graph intersects the x-axis at x = 3.

Statement 4, When x decreases, the function increases because the exponential term \(e^{3-x}\) becomes smaller and smaller as x decreases, which causes the entire expression 2 - \(e^{3-x}\) to increase.

Correct Question :

Write a function for which all four statements are true.

Statement 1: The domain of the function is x ≤ 3.

Statement 2: The range of the function is y ≥ 0.

Statement 3: The x-intercept of the function is 3.

Statement 4: When x decreases, the function increases.

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According to the quantity theory of money, changes in the money supply will lead directly to changes in the price level if velocity and real GDP are unaffected by the change in the money supply.Velocity can change over time, and changes in velocity may be caused by various factors.

For example, changes in velocity can be caused by shifts in payment practices, changes in the use of credit, changes in the availability of bank deposits or cash, or shifts in demand patterns.Changes in velocity may be related to changes in the money supply.

For example, if the money supply increases, the demand for money may increase, causing the velocity of money to decrease. Conversely, if the money supply decreases, the demand for money may decrease, causing the velocity of money to increase.

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The domain and range of the graph above in interval notation include the following:

Domain = [-6, 3]

Range = [-3, 3]

In Mathematics and Geometry, a domain refers to the set of all real numbers (x-values) for which a particular function (equation) is defined.

In Mathematics and Geometry, the horizontal portion of any graph is used to represent all domain values and they are both read and written from smaller to larger numerical values, which simply means from the left of any graph to the right.

By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:

Domain = [-6, 3] or -6 ≤ x < 3.

Range = [-3, 3] or -3 < y < 3

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Answer:

it is a decimal which is 3.28571428571

Step-by-step explanation:

i am in college

Step-by-step explanation:

\( \\ 9 \frac{6}{7} \div 3\)

Change the mixed fraction into improper fraction.

\( \\ = \frac{69}{7} \div 3\)

Reciprocal 3 for cross multiplication.

\( \\ = \frac{69}{7} \times \frac{1}{3} \)

Divide 69 by 3.

\( \\ = \frac{ \cancel{69}}{7} \times \frac{1}{ \cancel{3}} \)

\( \\ = \frac{23}{7} \)

Divide the result into mixed fraction or decimal.

\( \\ = 3.285 \: \: or \: \: 3 \frac{2}{7} \)

[Answer]

Graph 2

It is the graph with the line on the positive Y-Axis as the equation is ended with +3

A polynomial function f(x) of degree 3 with given zeros, 2, -3 , 5 and condition f(3)=6 is given by f(x) = -1/2 (x-2)(x+3)(x-5).

Degree of the polynomial function is equal to 3.

Zero's of the polynomial function is equal to 2, -3 , 5 .

And f(3) = 6

If 2, -3, and 5 are zeros of f(x), then we can write function as,

f(x) = a(x-2)(x+3)(x-5)

where a is some constant.

To determine the value of a, we can use the fact that f(3) = 6.

Substituting x = 3 into the equation above, we get,

f(3) = a(3-2)(3+3)(3-5)

     = -12a

Here we have,

f(3) = 6, so we can set these two expressions equal to each other and solve for a,

⇒ -12a = 6

⇒ a = -1/2

Now we can substitute this value of a back into the equation for f(x) that we found earlier,

⇒ f(x) = -1/2 (x-2)(x+3)(x-5)

Therefore, a polynomial function f(x) of degree 3 with zeros 2, -3, 5, and f(3) = 6 is equal to f(x) = -1/2 (x-2)(x+3)(x-5).

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Answer:

\(115 \dfrac{1}{2}\:cm^3\)

Step-by-step explanation:

The volume of a rectangular prism is the product of each of the sides of the prism

Given the sides have lengths
\(3\dfrac{1}{2}, 6 \;and\; 5 \dfrac{1}{2} cm\)

the volume would be
\(3\dfrac{1}{2} \times 6 \times 5 \dfrac{1}{2}\)

To perform this multiplication, convert mixed fractions to improper fractions first

Use the rule that mixed fraction
\(a\dfrac{b}{c}=\dfrac{a\times \:c+b}{c}\)

\(3\dfrac{1}{2}=\dfrac{3\times 2+1}{2} = \dfrac{7}{2}\)

\(5\dfrac{1}{2}=\dfrac{5\times 2+1}{2}= \dfrac{11}{2}\)

Therefore
\(3\dfrac{1}{2}\times \:6\times \:5\dfrac{1}{2}\\\\= \dfrac{7}{2}\times \:6\times \dfrac{11}{2}\\\\= \dfrac{7}{2}\times \dfrac{6}{1}\times \dfrac{11}{2} \quad(6 = \dfrac{6}{1})\)

\(=\dfrac{7\times \:6\times \:11}{2\times \:1\times \:2}\\\\= \dfrac{462}{4}\\\)

Divide numerator and denominator by 2 to get
\(\dfrac{231}{2}\\\)

Convert improper fraction \(\dfrac{231}{2}\) to mixed fraction using quotient/remainder

\(\dfrac{231}{2} \\\\\rightarrow Quotient: 115\\\\\rightarrow Remainder = 231 - 115 \times 2 = 231 - 230 = 1\)

\(\dfrac{231}{2} = 115 \dfrac{1}{2}\)

Answer:

add me as a friend and give me thanks

Step-by-step explanation:

the answer is A for sure

The confidence interval is 0.559 < p < 0.641 and expected value is 0.600

To construct a confidence interval for the true population proportion, we can use the formula:

p ± Z * √((p × (1 - p)) / n)

Where:

p = sample proportion (336/560)

Z = critical value for the desired confidence level

95% confidence = Z-value of approximately 1.96

n = 560

Let's calculate the confidence interval:

p = 336/560 ≈ 0.600

Z ≈ 1.96 (for a 95% confidence level)

n = 560

Plugging these values into the formula:

p ± Z × √((p × (1 - p)) / n)

0.600 ± 1.96 × √((0.600 × (1 - 0.600)) / 560)

0.600 ± 1.96 × √((0.240) / 560)

0.600 ± 1.96 × √(0.0004285714)

0.600 ± 1.96 × 0.020709611

0.600 ± 0.040564459

The confidence interval is:

0.559 < p < 0.641

Therefore, the 95% confidence interval for the true population proportion of adults with children is 0.559 < p < 0.641.

For proportions, the expected value is simply the sample proportion, which is approximately 0.600.

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Answer:

i=prt /100

200×2×9÷100=3.6