Suppose - 2x²-20x-51 ≤ f(x) ≤ 2x²+20x +49 for all x values near -5, except possibly at -5.
Evaluate lim f(x).
X→-5

Answer:

B

Step-by-step explanation:

calculate the slopes m of the 2 lines using the slope formula

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

with (x₁, y₁ ) = (- 6, 5 ) and (x₂, y₂ ) = (- 2, 7 )

m = [tex]\frac{7-5}{-2-(-6)}[/tex] = [tex]\frac{2}{-2+6}[/tex] = [tex]\frac{2}{4}[/tex] = [tex]\frac{1}{2}[/tex]

repeat with

(x₁, y₁ ) = (4, 2 ) and (x₂, y₂ ) = (6, 6 )

m = [tex]\frac{6-2}{6-4}[/tex] = [tex]\frac{4}{2}[/tex] = 2

• Parallel lines have equal slopes

[tex]\frac{1}{2}[/tex] ≠ 2 , then lines are not parallel

the product of the slopes of perpendicular lines equals - 1

[tex]\frac{1}{2}[/tex] × 2 = 1 ≠ - 1 , thus lines are not perpendicular

the lines are neither perpendicular nor parallel

Since two-way table shows the distribution of high school students The correct option is C: No, events A and B are not independent because P(A) ≠ P(A|B).

To determine whether events A and B are independent, we need to compare the probability of event A occurring without any knowledge of event B (i.e., P(A)) to the probability of event A occurring given that event B has occurred (i.e., P(A|B)).

From the table, we see that P(A) = 8/32 = 1/4, since there are 8 juniors out of a total of 32 students. Also, P(B) = 8/32 = 1/4, since there are 8 students who prefer earbuds out of a total of 32 students.

From the table, we see that the probability of both events A and B occurring is P(A and B) = 4/32 = 1/8, since there are 4 juniors who prefer earbuds out of a total of 32 students.

Now, to calculate P(A|B), we use the formula:

P(A|B) = P(A and B) / P(B)

Substituting the values we have, we get:

P(A|B) = (1/8) / (1/4) = 1/2

Since P(A) = 1/4 and P(A|B) = 1/2, we can see that P(A) ≠ P(A|B). Therefore, events A and B are not independent.

Therefore, the answer is No, P(A) ≠ P(A|B).

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Approximately 342,857 empty cans would weigh 200 pounds.

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other. For example, if there is 1 boy and 3 girls you could write the ratio as 1 : 3 (for every one boy there are 3 girls).

To solve this problem, we need to use proportional reasoning. We know that six empty cans weigh 3.5 ounces, so we can set up a proportion to find out how many cans are needed to weigh 200 pounds:

6 cans : 3.5 ounces = x cans : 200 pounds

To solve for x, we need to cross-multiply and simplify:

6 cans * 200 pounds = 3.5 ounces * x cans

1200 pounds = 3.5 ounces * x cans

x cans = 1200 pounds / 3.5 ounces

x cans = 342857.14 cans (rounded to two decimal places)

Therefore, approximately 342,857 empty cans would weigh 200 pounds.

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Therefore, the values of a and b are:

[tex]a = 2[/tex]

[tex]b = 9[/tex]

[tex]So, the functionf(x)=2x+9.[/tex]

In programming and mathematics, a function is a set of instructions or a block of code that performs a specific task or calculation. A function takes in one or more input values, performs a series of operations on them, and then returns a single output value. Functions are used to modularize code, making it easier to read, understand, and maintain. They can be defined and called within a program to perform repetitive tasks, manipulate data, or solve complex problems. Functions can also be used to create reusable code that can be called from multiple parts of a program or shared between different programs.

To find the values of a and b, we need to use the given information to create a system of two equations.

Let's start by using the first two rows of the table:

[tex]f(0) = a(0) + b = 7[/tex]

[tex]f(1) = a(1) + b = 9[/tex]

Simplifying these equations, we get:

[tex]b = 7\\a + b = 9[/tex]

Substituting b = 7 into the second equation, we get:

[tex]a + 7 = 9[/tex]

Solving for a, we get:

[tex]a = 2[/tex]

Now we can use this value of a to find b. Let's use the third and fourth rows of the table:

[tex]f(2) = a(2) + b = 11\\f(3) = a(3) + b = 13[/tex]

Substituting a = 2 into these equations, we get:

[tex]2 + b = 11\\3 + b = 13[/tex]

Solving for b, we get:

[tex]b = 9[/tex]

Therefore, the values of a and b are:

[tex]a = 2\\b = 9[/tex]

[tex]So, the functionf(x)=2x+9.[/tex]

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Answer:

-106f+56

Step-by-step explanation:

7(-8f-2)= -56f-14

10(-5f+7)= -50f+70

-56f-14. +. 50f+70

=-106f+56

Answer:

-2(53f-28)

Step-by-step explanation:

7(−8f−2)+10(−5f+7)

7x-8f=-56f  7x-2=-14

10x-5f=-50f   10x7=70

-56f-14+-50f+70       Adding a negative to something means to subtract. Like right here at -14 + -50 is the same as -14 - 50

   2. Add the numbers

-56f-14+-50f+70   -14+70=56

-56f+56-50f

   3. Combine like terms

-56f+56-50f   -56+ -50= -106

-106f +56

   4. Common Factor

-106f + 56

-2(53f-28)

The polynomial after simplification is obtained as [tex]5 x^{2}[/tex] + 9x + 3.

What is simplification?

In mathematics, reducing an expression, fraction, or problem to a simpler form is referred to as simplification. With calculations and solution, the problem is made simple. Make something simpler by simplifying it.

We are given a polynomial as

5x - 5 + [tex]11 x^{2}[/tex] + 4x - [tex]6 x^{2}[/tex] + 8

Now, in order to simplify the given polynomial, we will combine the like terms of the polynomial.

On combining the like terms, we get

[tex]5 x^{2}[/tex] + 9x + 3

Hence, the polynomial after simplification is obtained as [tex]5 x^{2}[/tex] + 9x + 3.

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The rate of change of the angle of elevation is 0.00011 rad/ft, for that we have to know the elevation.

The height above or below a specified(predefined) reference point, most frequently a reference geoid, which is a mathematical representation (expression) of the Earth's sea level as an equipotential gravitational surface, determines(defines) a location's elevation.

The angle of elevation of the camera is, Θ= [tex]tan^-^1(\frac{x}{2000} )[/tex]

The distance between the camera and the launching pad is, b= 2000 ft

The distance between the camera and the racket is, h = 4255 ft

To find the rate of change of the angle of elevation of the camera, we need to differentiate the function of that angle.

Θ= [tex]tan^-^1(\frac{x}{2000} )[/tex]

Differentiating function with respect to x we have,

[tex]\frac{d}{dx}[/tex] Θ=[tex]\frac{d}{dx} tan^-^1(\frac{x}{2000} )[/tex]

Θ'=[tex]\frac{1}{1+\frac{x^2}{2000^2} }[/tex][tex]*\frac{1}{2000}[/tex]

[tex]\frac{1}{2000+\frac{x^2}{2000} }[/tex]  (Equation 1)

[tex]h^2=b^2+p^2[/tex]

[tex]4255^2+2000^2+x^2[/tex]

Θ=[tex]\frac{1}{2000+\frac{14105025}{2000} }[/tex]

≈ 0.00011 rad/ft.

Thus, the rate of change of the angle of elevation is 0.00011 rad/ft.

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According to the given information, it takes 0.5 hours to travel 16.5 highway miles and 7 city miles and 1.6 hours to travel 44 highway miles and 28 city miles.

What is the average rate?

The average rate is a measure of the average speed at which something occurs over a period of time. It is calculated by dividing the total amount of something by the total time it took to occur.

The formula for the average rate or average speed is:

Average speed = Total distance traveled / Total time taken

a. To find the time it takes to travel 16.5 highway miles and 7 city miles, we can use the formula:

time = distance/speed

The time it takes to travel 16.5 highway miles is:

time on highway = 16.5 / 55 = 0.3 hours

The time it takes to travel 7 city miles is:

time in city = 7 / 35 = 0.2 hours

The total time it takes to travel both distances is:

total time = time on highway + time in city = 0.3 + 0.2 = 0.5 hours

Therefore, it takes 0.5 hours (or 30 minutes) to travel 16.5 highway miles and 7 city miles.

b. To find the time it takes to travel 44 highway miles and 28 city miles, we can use the same formula:

time on highway = 44 / 55 = 0.8 hours

time in city = 28 / 35 = 0.8 hours

total time = time on highway + time in city = 0.8 + 0.8 = 1.6 hours

Therefore, it takes 1.6 hours (or 96 minutes) to travel 44 highway miles and 28 city miles.

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The single translation that maps shape A onto shape C is given as follows:

A translation happens when either a figure or a function are moved horizontally or vertically on the coordinate plane.

The four translation rules for coordinates are defined as follows:

Translations can often be composed, that is, a combination of multiple translations can result in a single translation, as is the case for this problem.

Considering the translation of 3 units left(shape B) and then 4 units right(shape C), the horizontal translation is given as follows:

-3 + 4 = 1 -> 1 unit right.

Considering the translation of 7 units down(shape B) and then 3 units down(shape C), the vertical translation is given as follows:

-7 - 3 = 10 units down.

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Answer:

Step-by-step explanation:

A hands-on experience is the only modality that would give visitors a complete understanding of the plight of modern enslaved people.

Answer:

Step-by-step explanation:

To find P(x<4), we need to use the cumulative distribution function (CDF) of the binomial distribution.

For N = 6 and p = 0.3, the probability mass function (PMF) is:

P(X = k) = (6 choose k) * 0.3^k * 0.7^(6-k)

where (6 choose k) is the binomial coefficient.

Using this formula, we can find the probabilities for each value of X less than 4:

P(X = 0) = (6 choose 0) * 0.3^0 * 0.7^6 ≈ 0.1176

P(X = 1) = (6 choose 1) * 0.3^1 * 0.7^5 ≈ 0.3025

P(X = 2) = (6 choose 2) * 0.3^2 * 0.7^4 ≈ 0.3241

P(X = 3) = (6 choose 3) * 0.3^3 * 0.7^3 ≈ 0.1852

Therefore, P(x<4) is the sum of these probabilities:

P(x<4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) ≈ 0.9294

So the probability of getting less than 4 successes in 6 trials with a success probability of 0.3 is approximately 0.9294.

Answer:

  (a)  5.62 m/s²

  (b)  22.5 m/s

Step-by-step explanation:

You want the magnitude of the acceleration of a crate on a frictionless plane inclined at 35° from the horizontal. And you want the speed after 4 seconds.

The acceleration will be the component of the acceleration due to gravity that is in the direction down the plane.

  (9.8 m/s²)·sin(35°) ≈ 5.62 m/s²

The speed of the crate will be the product of the acceleration and time:

  v = at = (5.6 m/s²)(4 s) = 22.5 m/s

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The prοduct οf the matrices is nοt the identity matrix. Therefοre, X and A nοt inverses οf each οther.

An identity matrix is a square matrix in which all the elements οf principal diagοnals are οne, and all οther elements are zerοs. It is denοted by the nοtatiοn “In” οr simply “I”. Any matrix will yield a matrix as a result of being multiplied by the identity matrix.

The identity matrix is a matrix having entry οne in its diagοnal and rest οf the entries are zerο.

[tex]X.A = \left[\begin{array}{cc}2&-1\\6&-4\\\end{array}\right] \left[\begin{array}{cc}1/2&1/4&\\-2&-1/2&\end{array}\right][/tex]

[tex]X.A = \left[\begin{array}{cc}3&1\\11&7/2\end{array}\right][/tex]

Hence, The prοduct of the matrices is not the identity matrix. Therefοre, X and A not inverses of each other.

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The complete question is in the attached image.

a. The Vοlume = 198 cm³ and height = 7, if it’s a cylinder with a radius οf 3cm.

b. The cοst οf cοffee is $3.96 when $0.02 per cubic centimetre.

The fοrmula fοr the vοlume οf a cylinder is

V = πr²h,

where V is the vοlume, r is the radius, and h is the height.

a) Tο find the height οf the cylinder, we need tο knοw the vοlume οf the cοntainer and the radius. The fοrmula fοr the vοlume οf a cylinder is:

V = πr²h

[tex]$\mathrm{\frac{V }{ h}} = \pi r\² }[/tex]

[tex]$\mathrm{\frac{V }{ h}} = \frac{22}{7} \times 3 \times 3}[/tex]

[tex]$\mathrm{\frac{V }{ h}} = \frac{22}{7} \times 3 \times 3}[/tex]

[tex]$\mathrm{\frac{V }{ h}} = \frac{198}{7}[/tex]

Thus, Vοlume = 198 cm³ and height = 7

b) If the cοntainer frοm is filled with cοffee and cοffee cοsts $0.02 per cubic centimetre, Then the cοst οf cοffee

= $0.02 × 198 cm³

= $3.96

Thus, The Vοlume = 198 cm³ and height = 7, if it’s a cylinder with a radius οf 3cm.

The cοst οf cοffee is $3.96 when $0.02 per cubic centimetre.

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Therefore, we have proven that cos (0) - sin (0) = √2 sin (0), given that sin (0) + cos (0) = √2 cos (0).

Trigonometry is a branch of mathematics that deals with the study of the relationships between the angles and sides of triangles. It explores the properties and functions of angles, as well as the relationships between them in various geometrical shapes.

Given by the question.

We have:

sin (0) + cos (0) = √2 cos (0)

Simplifying the left-hand side:

sin (0) + cos (0) = 1

Substituting back into the original equation:

1 = √2 cos (0)

Dividing both sides by √2:

1/√2 = cos (0)

Now we need to prove that:

cos (0) - sin (0) = √2 sin (0)

Substituting our value for cos (0):

1/√2 - sin (0) = √2 sin (0)

Multiplying both sides by √2:

√2/2 - √2 sin (0) = 2 sin (0)

Adding √2 sin (0) to both sides:

√2/2 = 3 sin (0)

Dividing both sides by 3:

sin (0) = √2/6

Substituting back into the equation we were trying to prove:

cos (0) - sin (0) = cos (0) - √2/6

We can now use our earlier equation to substitute for cos (0):

cos (0) - √2/6 = √2/2 - √2/6

Simplifying:

cos (0) - sin (0) = √2 sin (0)

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The six trig functions are:

(1) Figure 1, sin θ = 1/√5

(2) Figure 1, cos θ = 2/√5

(3) Figure 1, tan θ = 1/2

(4) Figure 2, sin θ = 3/√13

(5) Figure 2, cos θ = 2/√13

(6) Figure 2, tan θ = 3/2

Trig function (short for "trigonometric function") is a mathematical function that relates the angles of a right triangle to the lengths of its sides. The main trig functions are sine (sin), cosine (cos),  and tangent (tan).

The trig functions are abbreviated using SOH CAH TOA.

SOH : sin θ = opposite / hypotheses

CAH: cos θ = adjacent / hypothenuse

TOA : tan θ = opposite / adjacent

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The relationship which describe angles 1 and 2 is supplementary angles.

The correct answer choice is option A.

Supplementary angles are angles in which the sum of their measure is equal to 180°.

Based on this, angles 1 and 2 are supplementary because they both add up to 180°.

Complementary angles are two angles which add up to 90 degrees.

Vertical angles are angles opposite each other when two angles intersect.

Adjacent angles are angles which have a common side and common vertex.

Hence, angles 1 and 2 are supplementary.

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Therefore , the solution of the given problem of slope comes out to be  y = 7x - 17 is the equation in slope-intercept form.

A line's steepness is determined by its slope. In gradient-based equations, a situation known as gradient overflow can happen. One can determine the slope by dividing the sum of a run (width differential) and rise (climbing distinction) between two places. The equation with the hill variation, y = mx + b, is used to model the fixed path issue. where grade is mm, a = bc, and the line's y-intercept is situated; (0, b).

Here,

Given :

=> -7x + y = -17

7x both faces together

=>  -7x + 7x + y = 7x - 17

Condense: y = 7x - 17

So,

=>  y = 7x - 17 is the equation in slope-intercept form.

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By the given definition of matrix A, the value of the matrix A⁻¹ is [tex]\left[\begin{array}{ccc}-1&0&1&2&-1&-2&0&0&-1\end{array}\right][/tex] and it is true that AA⁻¹ = I₃.

Given that

[tex]A = \left[\begin{array}{ccc}1&0&2\\1&1&0\\1&0&1\end{array}\right][/tex]

To find the inverse of A, we can use the following formula:

A⁻¹ = (1/|A|) adj(A)

where |A| is the determinant of A and adj(A) is the adjugate matrix of A.

First, let's calculate the determinant of A:

|A| = 1(1×1 - 0×0) - 0(1×0 - 0×1) + 2(1×0 - 1×1)

|A| = -1

Next, we need to find the adjugate matrix of A.

To do this, we need to find the matrix of cofactors of A, then take its transpose:

[tex]C = \left[\begin{array}{ccc}1&-2&0&0&1&0&-1&2&1\end{array}\right]\\[/tex]

adj(A) = [tex]C^T = \left[\begin{array}{ccc}1&0&-1&-2&1&2&0&0&1\end{array}\right][/tex]

Now we can calculate A⁻¹ using the formula:

A⁻¹ = (1/|A|) adj(A)

So, we have

[tex]A^{-1} = (-1) \left[\begin{array}{ccc}1&0&-1&-2&1&2&0&0&1\end{array}\right] = \left[\begin{array}{ccc}-1&0&1&2&-1&-2&0&0&-1\end{array}\right][/tex]

Finally, we can check whether AA⁻¹ = I₃ using

[tex]A^{-1}A = \left[\begin{array}{ccc}-1&0&1&2&-1&-2&0&0&-1\end{array}\right] \left[\begin{array}{ccc}1&0&2&1&1&0&1&0&1\end{array}\right] = \left[\begin{array}{ccc}1&0&0&0&1&0&0&0&1\end{array}\right] = I_3[/tex]

Therefore, we have found the inverse of A and verified that AA⁻¹ = I₃.

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Perimeter of the given polygon figure will be 38 units.

What exactly is a polygon in mathematics?

A closed, two-dimensional, flat, closed polygon is a shape that is constrained by geometry and has straight sides. Its sides don't curve inward at all. Another term for a polygon's sides is its polygonal edges. The points where two sides converge are known as a polygon's vertices (or corners).

Perimeter of a polygon = Measure of the circumference of the polygon

From the figure attached,

Perimeter of the figure = measure of the three straight lines + measure of the circumference of the quarter circle

Measure of linear sides = 5 + 10 + 10 + 5 = 30 units

Circumference of the quarter circle = 1/4(2πr)

                                                          =  1/2(π)(5)

                                                          = 7.85 units

Perimeter of the figure = 30 + 7.85

                                     = 37.85

                                     ≈ 38 units

  Therefore, perimeter of the given figure will be 38 units.

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Answer:

Step-by-step explanation:

Logically, if a polynomial of real coefficients has a complex root, it’s conjugate is a root as well. That means if -20-7i is a root, -20+7i is also a root. Also, since we’re dealing with minimal degree, we wish to construct a quadratic expression with such roots. To do this, we can use Vieta’s formulas, which relates the trinomial coefficients with its roots. The two roots added together is -b/a in ax^2 + bx + c, and the two roots multiplied together is c/a. The two roots added together is -40. The two roots multiplied together is 449. So, our quadratic expression is x^2+40x+449

It will run dry after 141.7 days.

The rainwater tank has a diameter of 160 cm and a height of 190 cm, then its volume is:

V = 3.14*(180cm)*(160cm/2)² = 3,617,280 cm³

We know that 1 L = 1000 cm³

then:

3,617,280 cm³ = 3,617.280 L

The water is used at a rate of 25,5 litres per day, so after x days there is:

W =  3,617.280  - 25.5*x liters

It is zero when:

0 = 3,617.280  - 25.5*x

x =  3,617.280/25.5

x = 141.7 days.

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Answer: 204 ft

Step-by-step explanation: First lets find the area of the triangle, the formula for area of a triangle is base times height divided by 2. Now the height isn't given but we can find it by subtracting 18 by 10 which gives us 8. 8 times 6 is 48 divided by 2 is 24. Therefore, the area of the triangle is 24 ft. Now we find the area of the rectangle which is length times width. 18 times 10 is 180. Now we add the area of both shapes, 180 plus 24 is 204, so your answer is 204 ft.

Answer:

Step-by-step explanation:

Using Law of sines, the missing angles are:

4) ∠B = 45.74°

6) ∠C = 21.02°

8) ∠A = 60.84°

The law of sines is a rule for solving the lengths and angles of triangles and it states that:

a/sin A = b/sin B = c/sin C

4) Using law of sines, we have:

8/sin B = 11/ sin 80

(8/11) sin 80 = sin B

sin B = 0.7162

B = sin⁻¹0.7162

∠B = 45.74°

6) Using law of sines, we have:

13/sin C = 32/sin 62

sin C = (13/32) sin 62

sin C = 0.3587

C = sin⁻¹0.3587

C = 21.02°

8) Using law of sines, we have:

30/sin 104 = 27/sin A

sin A = (27/30) sin 104

sin A = 0.8733

A = sin⁻¹0.8733

A = 60.84°

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Answer:

x = 16

Step-by-step explanation:

∠1 + ∠2 = 90

42 + 3x = 90

3x = 90 - 42

x = 48/3 = 16

Therefore, the equilibrium level of national income in this basic Keynesian macroeconomic model is 480.

In mathematics, the sum is the result of adding two or more numbers or quantities together. The symbol used to represent a sum is the capital Greek letter sigma (∑), which stands for "sum up."

by the question.

In the basic Keynesian macroeconomic model, equilibrium occurs when national income (Y) equals total spending in the economy. Therefore, we can set Y equal to the sum of consumption (C) and investment (I), as follows:

Y = C + I

Substituting the given consumption and investment functions into this equation, we get:

Y = (40 + 0.5Y) + 200

Simplifying and rearranging, we get:

Y - 0.5Y = 240

0.5Y = 240

Y = 480

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If the value of x is 4, the quadrilateral meets all the requirements of a parallelogram and will be a parallelogram.

A quadrilateral is a four-sided shape with straight line segments as its sides. It is a two-dimensional shape with four angles, each measuring 90°. Quadrilaterals can be categorized into different shapes such as squares, rectangles, parallelograms, trapezoids, rhombuses, and kites.

A parallelogram is a four-sided shape with opposite sides parallel and equal in length. To find the value of x that makes the quadrilateral a parallelogram, we need to use the property of a parallelogram, which states that the opposite sides of a parallelogram are equal in length.

Therefore, we need to equate the opposite sides of the quadrilateral: 6x-8=4x. After solving this equation, we get x=4. Thus, the value of x that makes the quadrilateral a parallelogram is 4.

In order for a quadrilateral to be a parallelogram, opposite sides must be parallel. Using this property, we can set the opposite sides equal to each other and solve for x.

[tex]$2x + 5 = 3x - 2$[/tex]

Simplifying this equation, we get:

[tex]$x = 7$[/tex]

Therefore, if [tex]$x=7$[/tex], the quadrilateral is a parallelogram.

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