Solve the inequality for x and identify the graph of its solution.
|x+31>2
Choose the answer that gives both the correct solution and the correct graph.
O A. Solution: x>1 and x < 5
-
-2 -1 0 1 2
2 3 4 5 6 7 8
OB. Solution: x < -5 or x>-1
OT
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
OC. Solution: x < -5 or x>-1
+
-O
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
OD. Solution: x>-5 and x < -1
O
-8-7-6-5-4-3 -2 -1 0 1 2

How to convert 18.8 hm (hectometers) to meters using the diagram. So, 18.8 hm is equal to 1,880 m after converting using the diagram.

Here is how to convert 18.8 hm to m using the diagram:

1. Start with the given measurement, which is 18.8 hm.

2. Locate "hecto" on the diagram and see that it is 2 units away from "m" (meter).

3. Since you want to convert from hm to m, you need to move 2 units to the left on the diagram.

4. Moving 2 units to the left on the diagram brings you to the meter (m) unit.

5. Each time you move one unit to the left on the diagram, you divide the original measurement by 10. Since you moved 2 units to the left, you need to divide 18.8 by 100 (10 raised to the second power).6. Therefore, 18.8 hm is equal to 1,880 m after converting using the diagram.

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The equation of the linear function is y = 4x - 2. The missing value in the table is 14.

The missing value in the given table is 10. Assuming that the data represent a linear function, we can use the slope formula (m = (y2 - y1) / (x2 - x1)) to find the missing value.

We can choose any two sets of coordinates from the table to find the slope, then use that slope to find the missing value. For example, using the coordinates (1, 2) and (2, 6), we get:m = (6 - 2) / (2 - 1) = 4

Therefore, the equation of the linear function is y = 4x + b, where b is the y-intercept. To find b, we can plug in any set of coordinates into the equation and solve for b.

For example, using the coordinates (1, 2), we get:2 = 4(1) + bSimplifying the equation, we get:b = -2Therefore, the equation of the linear function is y = 4x - 2. We can now use this equation to find the missing value when x = 4:y = 4(4) - 2 = 14Therefore, the missing value in the table is 14.

We are given a table of x and y values. We need to find the missing value assuming that the data represent a linear function. To find the missing value, we can use the slope formula (m = (y2 - y1) / (x2 - x1)).

This formula gives us the slope of the line passing through two points, which is the rate at which y changes with respect to x. We can then use this slope to find the missing value in the table.
Using the coordinates (1, 2) and (2, 6), we get a slope of 4. This means that for every 1 unit increase in x, y increases by 4 units. Therefore, the equation of the linear function is y = 4x + b, where b is the y-intercept.

To find b, we can plug in any set of coordinates into the equation and solve for b. Using the coordinates (1, 2), we get b = -2. Therefore, the equation of the linear function is y = 4x - 2.
Finally, we can use this equation to find the missing value when x = 4. Plugging in x = 4, we get y = 4(4) - 2 = 14. Therefore, the missing value in the table is 14.

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Answer:

B. 2 and 4 es la respuesta correcta

The expression [tex](m+2z)^2+12tz[/tex] simplifies to [tex]m^2 + 4mz + 4z^2 + 12tz[/tex]

The expression [tex](m+2z)^2+12tz[/tex] represents a mathematical equation involving variables m and z, as well as the constant t.

To simplify the expression, we can expand the square and then combine like terms.

Expanding the square, we have:  

[tex](m+2z)^2 = (m+2z)(m+2z) = m^2 + 4mz + 4z^2[/tex]

Substituting this result back into the original expression, we have:

[tex](m+2z)^2 + 12tz = m^2 + 4mz + 4z^2 + 12tz[/tex]

At this point, we have combined all the terms in the expression, and there are no more like terms to be simplified.

Therefore, the final simplified form of the expression [tex](m+2z)^2+12tz is m^2 + 4mz + 4z^2 + 12tz.[/tex]

It is important to note that this simplified expression is still in terms of the original variables m, z, and t, and no further simplification can be done unless specific values are assigned to these variables.

This equation can be further manipulated or solved depending on the context or purpose it serves within a mathematical problem or equation system.

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The correct statement regarding the graph of the linear function (g - f)(x) is given as follows:

On a coordinate plane, a straight line with a positive slope goes through points (negative 2, negative 6), (0, 0), and (2, 6).

The function f(x) in this problem is defined as follows:

f(x) = -x.

The function g(x) in this problem is defined as follows:

g(x) = 2x.

Hence the subtraction of the two functions is given as follows:

(g - f)(x) = g(x) - f(x)

(g - f)(x) = 2x - (-x)

(g - f)(x) = 3x

Which has a positive slope, hence the correct option is given as follows:

On a coordinate plane, a straight line with a positive slope goes through points (negative 2, negative 6), (0, 0), and (2, 6).

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Answer:

To find the square root of 0.327 up to 2 decimal places, you can use a calculator or mathematical software. The square root of 0.327 is approximately 0.57.

The value of the constant k in the probability mass function is 1/4, and P(Z < 3) is equal to 37/64.

To find the value of the constant k in the probability mass function (PMF) and calculate P(Z < 3), we can use the properties of a discrete random variable and the given PMF.

First, we know that the sum of probabilities for all possible values of a discrete random variable must equal 1. Therefore, we can write:

∑ P(Z = z) = 1

Now let's substitute the given PMF into the summation:

∑ k[tex](3/4)^z[/tex] = 1

We can simplify this expression by factoring out the constant k:

k ∑ [tex](3/4)^z[/tex] = 1

Next, we need to evaluate the summation term. The summation represents a geometric series with a common ratio of 3/4. The sum of a geometric series is given by:

∑[tex]r^n[/tex] = 1 / (1 - r), where |r| < 1

In this case, the summation term becomes:

∑ [tex](3/4)^z[/tex] = 1 / (1 - 3/4)

Simplifying further:

∑ [tex](3/4)^z[/tex] = 1 / (1/4) = 4

Now, we can substitute this value back into the previous equation:

k * 4 = 1

Solving for k:

k = 1/4

Therefore, the value of the constant k is 1/4.

Now let's calculate P(Z < 3) using the PMF:

P(Z < 3) = P(Z = 0) + P(Z = 1) + P(Z = 2)

Substituting the given PMF with k = 1/4:

P(Z < 3) = (1/4)(3/4)^0 + (1/4)(3/4)^1 + (1/4)(3/4)^2

        = (1/4)(1) + (1/4)(3/4) + (1/4)(9/16)

        = 1/4 + 3/16 + 9/64

        = 16/64 + 12/64 + 9/64

        = 37/64

Therefore, P(Z < 3) is equal to 37/64.

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The expression 2m(3/2m + 1) + 3(5/3m - 2) is equivalent to 11m/3 - 3.

To simplify the given expression, let's distribute and combine like terms:

2m(3/2m + 1) + 3(5/3m - 2)

First, distribute 2m to the terms inside the parentheses:

= (2m * 3/2m) + (2m * 1) + (3 * 5/3m) - (3 * 2)

Simplifying each term:

= (3) + (2m) + (5m/3) - 6

Next, combine like terms:

= 2m + (5m/3) - 3

To add the terms 2m and (5m/3), we need a common denominator.

The common denominator is 3:

= (6m/3) + (5m/3) - 3

Combining the terms with the common denominator:

= (6m + 5m)/3 - 3

= 11m/3 - 3

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The area of the rhombus with a base of 18 centimeters and a height of 15.5 centimeters is 139.5 cm² (option c).

To find the area of a rhombus, we can use the formula:

Area = (base * height) / 2.

Given that the base of the rhombus is 18 centimeters and the height is 15.5 centimeters, we can substitute these values into the formula:

Area = (18 cm * 15.5 cm) / 2.

Multiplying the base and height gives us:

Area = (279 cm²) / 2.

Dividing 279 cm² by 2 gives us the final area:

Area = 139.5 cm².

Therefore, the area of the rhombus is 139.5 cm².

Thus, the correct option is c.

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Area: 11.18 sq units

Perimeter: 26.18 units

To find the area and perimeter of the triangle with vertices A(-6, -1), B(-3, 3), and C(5, -3), follow these steps:

1. Find the lengths of the sides:

  - Side AB: Calculate the distance between points A and B using the distance formula: AB = √[(x₂ - x₁)² + (y₂ - y₁)²]

    Substitute the coordinates: AB = √[(-3 - (-6))² + (3 - (-1))²] = √[3² + 4²] = √(9 + 16) = √25 = 5 units

  - Side BC: Calculate the distance between points B and C using the distance formula: BC = √[(x₂ - x₁)² + (y₂ - y₁)²]

    Substitute the coordinates: BC = √[(5 - (-3))² + (-3 - 3)²] = √[8² + (-6)²] = √(64 + 36) = √100 = 10 units

  - Side CA: Calculate the distance between points C and A using the distance formula: CA = √[(x₂ - x₁)² + (y₂ - y₁)²]

    Substitute the coordinates: CA = √[(-6 - 5)² + (-1 - (-3))²] = √[(-11)² + 2²] = √(121 + 4) = √125 = 11.18 units

2. Calculate the perimeter:

  Perimeter = AB + BC + CA = 5 + 10 + 11.18 = 26.18 units

3. Find the area using the Shoelace Formula:

  Area = 0.5 * |(x₁y₂ + x₂y₃ + x₃y₁) - (x₂y₁ + x₃y₂ + x₁y₃)|

  Substitute the coordinates: Area = 0.5 * |(-6 * 3 + (-3) * (-3) + 5 * (-1)) - (-3 * (-1) + 5 * 3 + (-6) * (-3))|

  Simplify: Area = 0.5 * |(-18 + 9 - 5) - (3 + 15 + 18)|

           = 0.5 * |-14 - 36|

           = 0.5 * |-50|

           = 0.5 * 50

           = 25 sq units

Therefore, the area of the triangle is 25 square units and the perimeter is 26.18 units.

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The system of equations given in option 4, y = 2x and 3x + 2y = 21, will have no solution.

2x + y = 9

2x + y = 5

The graph is attached

A system of linear equations will have no solution if the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts.

Looking at the given options:

y = 12 – 3x

y = 2x – 3

y = x – 1

-5x + y = -5

2x + y = 9

2x + y = 5

y = 2x

3x + 2y = 21

Option 3, 2x + y = 9 and 2x + y = 5, represents parallel lines.

The slopes of the lines are the same (both equations have a coefficient of -2 for x and 1 for y), but the y-intercepts are different

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Answer:

Circumference = Diameter × π

Circumference = 35 cm × 3.14

Circumference = 109.9 cm

The number of revolutions in 1 minute is given as 100, so the distance covered in 1 minute can be calculated as:

Distance in 1 minute = Circumference × Revolutions

Distance in 1 minute = 109.9 cm × 100

Distance in 1 minute = 10990 cm

To find the distance covered in half an hour, we need to multiply the distance in 1 minute by 30, since there are 30 minutes in half an hour:

Distance in half an hour = Distance in 1 minute × 30

Distance in half an hour = 10990 cm × 30

Distance in half an hour = 329700 cm

To express the answer in km, we need to divide the distance in cm by 100000, since there are 100000 cm in a km:

Distance in km = Distance in cm / 100000

Distance in km = 329700 cm / 100000

Distance in km = 3.297 km

Therefore the distance covered by the wheel in half an hour is 3.297 km.

If α is an ordinal and β ∈ α, then β satisfies all three properties of an ordinal. Therefore, β is also an ordinal.

To prove the statement "If α is an ordinal and β ∈ α, then β is an ordinal," we need to demonstrate that if α is an ordinal and β is an element of α, then β satisfies the three properties of an ordinal:

Well-Ordering: Every element of β is strictly well-ordered by the membership relation ∈. This property holds because α is an ordinal and satisfies the well-ordering property, and β being an element of α inherits this property.

Transitivity: For any two elements γ and δ in β, if γ ∈ δ and δ ∈ β, then γ ∈ β. Since β is an element of α and α is transitive, the transitivity property carries over to β.

Trichotomy: For any two elements γ and δ in β, either γ ∈ δ, δ ∈ γ, or γ = δ. Again, this property is inherited from α, as β is an element of α.

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Answer:

We can use the information that 24 families watched 16 hours or less to estimate the sample size. Since the 7th percentile is 16 hours, we know that 7% of the sample watched 16 hours or less. Therefore, we can set up a proportion:

(24 / x) = 0.07

where x is the total number of families in the sample. Solving for x, we get:

x = 24 / 0.07 ≈ 343

Rounding to the nearest integer, we can estimate that there are approximately 343 families in the sample.

Answer: p=48.8372093%

Step-by-step explanation:

Step-by-step explanation:

100% = 43

1% = 100%/100 = 43/100 = 0.43

21 is that many % as times 1% fits into 21 :

p = 21 / 0.43 = 48.8372093... %

The composite function for this problem is given as follows:

16.

The composite function of g(x) and f(x) is defined by the function presented as follows:

(g ∘ f)(x) = g(f(x)).

For the composition of two functions, we have that the output of the inner function, which in this example is given by g(x), serves as the input of the outer function, which in this example is given by f(x).

The functions for this problem are given as follows:

Hence the composite function is given as follows:

(g ∘ f)(x) = g(f(x)) = g(4) = 5(4) - 4 = 20 - 4 = 16.

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Answer:

when y=0

2x=12

we divide by 2 on both sides

x= 6

The simplified ratio of blue dye drops to the total drops of dye is 2:5. This means that for every 2 drops of blue dye, there are 5 drops of dye in total.

To find the ratio of drops of blue dye to the total drops of dye, we need to determine the total number of drops of dye used. Julian used 18 drops of yellow dye and 12 drops of blue dye, which gives us a total of 18 + 12 = 30 drops of dye.

Now, let's find the ratio of blue dye drops to the total drops of dye. The ratio can be expressed as "blue dye drops: total drops of dye." In this case, the number of blue dye drops is 12, and the total number of drops of dye is 30.

To simplify the ratio, we can divide both numbers by their greatest common divisor (GCD). The GCD of 12 and 30 is 6. Dividing both numbers by 6, we get 12 ÷ 6 = 2 and 30 ÷ 6 = 5.

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The factored form of the expression 81m^4 - n^2 is (9m^2 + n)(9m^2 - n).

To factor the expression 81m^4 - n^2, we need to look for common factors and apply algebraic factoring techniques.

We start by recognizing that 81m^4 can be written as (9m^2)^2, which is a perfect square. Similarly, n^2 is also a perfect square.

Applying the difference of squares formula, we can factor the expression as follows:

81m^4 - n^2 = (9m^2)^2 - n^2

             = (9m^2 + n)(9m^2 - n)

Now, the expression is factored in terms of (9m^2 + n) and (9m^2 - n). These factors represent the difference of two squares, where (9m^2 + n) is the sum of two squares (9m^2 and n^2) and (9m^2 - n) is the difference of two squares.

The factored form of the expression 81m^4 - n^2 is (9m^2 + n)(9m^2 - n).

Factoring expressions can be a useful technique to simplify and analyze mathematical expressions. In this case, we were able to factor the expression by recognizing perfect square terms and applying the difference of squares formula.

Factoring can help in solving equations, simplifying expressions, and identifying important patterns and relationships in mathematics.

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The expression for their distance apart d at the end of t hours is sqrt[tex](t^2 * [(car's speed)^2 + (truck's speed)^2][/tex]), and the exact value for t can be found using t = sqrt([tex]2500 / [(car's speed)^2 + (truck's speed)^2])[/tex].

To find an expression for the distance apart between the car and the truck at the end of t hours, we can use the concept of distance traveled. Since the car is heading east and the truck is heading south, the distances traveled by each vehicle can be represented as follows:

Distance traveled by the car = (car's speed) * (time) = t * (car's speed)

Distance traveled by the truck = (truck's speed) * (time) = t * (truck's speed)

To visualize the distance between the car and the truck, we can form a right triangle with the car's distance traveled as the horizontal leg and the truck's distance traveled as the vertical leg. The distance between them, represented by d, can be calculated using the Pythagorean theorem:

[tex]d^2 = (car's distance traveled)^2 + (truck's distance traveled)^2[/tex]

Substituting the expressions for the distances traveled:

[tex]d^2 = (t * (car's speed))^2 + (t * (truck's speed))^2[/tex]

Simplifying:

[tex]d^2 = t^2 * [(car's speed)^2 + (truck's speed)^2][/tex]

Taking the square root of both sides to find the distance d:

d = sqrt[tex](t^2 * [(car's speed)^2 + (truck's speed)^2])[/tex]

Therefore, the expression for the distance apart between the car and the truck at the end of t hours is:

d = sqrt[tex](t^2 * [(car's speed)^2 + (truck's speed)^2])[/tex]

Now, if at the end of t hours the two vehicles are 50 miles apart, we can set the expression equal to 50 and solve for t:

50 = sqrt([tex]t^2 * [(car's speed)^2 + (truck's speed)^2])[/tex]

Squaring both sides:

[tex]2500 = t^2 * [(car's speed)^2 + (truck's speed)^2][/tex]

Dividing by [(car's speed)^2 + (truck's speed)^2]:

[tex]t^2 = 2500 / [(car's speed)^2 + (truck's speed)^2][/tex]

Taking the square root:

t = sqrt[tex](2500 / [(car's speed)^2 + (truck's speed)^2])[/tex]

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At the end of 1 year, the amount in the account is $10,640, and at the end of 2 years, the amount is $11,910.40.

To calculate the amount in the account at the end of each year, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = the final amount

P = the principal amount (initial deposit)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the number of years

In this case, Boris placed $9500 in an account that pays 12% interest compounded annually.

(a) To find the amount in the account at the end of 1 year, we have:

P = $9500

r = 12% = 0.12

n = 1 (compounded annually)

t = 1 year

Using the formula, we have:

A = 9500(1 + 0.12/1)^(1*1)

A = 9500(1 + 0.12)^1

A = 9500(1.12)

A = $10640

Therefore, the amount in the account at the end of 1 year is $10,640.

(b) To find the amount in the account at the end of 2 years, we have:

P = $9500

r = 12% = 0.12

n = 1 (compounded annually)

t = 2 years

Using the formula, we have:

A = 9500(1 + 0.12/1)^(1*2)

A = 9500(1 + 0.12)^2

A = 9500(1.12)^2

A = $11910.40

Therefore, the amount in the account at the end of 2 years is $11,910.40.

In summary, at the end of 1 year, the amount in the account is $10,640, and at the end of 2 years, the amount is $11,910.40.

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Answer:

The correct answer is C. No. One output value has more than one input value. A relation is a function if and only if each input value corresponds to exactly one output value. In this case, the output value 14 corresponds to two different input values, 8 and 17, so the relation is not a function.

The visual proof demonstrates the Pythagorean Theorem by showing how the areas of squares constructed on the sides of a right triangle relate to each other.

In a visual proof of the Pythagorean Theorem, different geometric figures are used to demonstrate the relationship between the squares of the sides of a right triangle. The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

To demonstrate this visually, various transformations and rearrangements of geometric figures are performed. The figures labeled 5 through 9 likely represent different squares or triangles that are used in the proof.

In this visual proof, figure 5 could potentially represent a square constructed on one side of the right triangle, figure 6 could represent a square constructed on the other side of the triangle, and figure 7 could represent a square constructed on the hypotenuse. The lengths of these squares or the areas they cover are related to the original triangle in image one.

The relationship between these figures and the Pythagorean Theorem is that the area of the square constructed on the hypotenuse (figure 7) is equal to the sum of the areas of the squares constructed on the other two sides (figures 5 and 6). This visually represents the mathematical equation a^2 + b^2 = c^2, where 'a' and 'b' are the lengths of the legs of the right triangle, and 'c' is the length of the hypotenuse.

Overall, the visual proof demonstrates the Pythagorean Theorem by showing how the areas of squares constructed on the sides of a right triangle relate to each other. By visually observing the transformations and arrangements of these figures, one can understand and verify the Pythagorean Theorem geometrically.

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Answer:

Since arc ACD is 326 degrees and arc AD is a semicircle (180 degrees), then arc ACB is:

arc ACB = arc ACD - arc AD

arc ACB = 326 - 180

arc ACB = 146 degrees

Since arc ACB is a central angle, it is equal to twice the inscribed angle BCA:

2 * BCA = arc ACB

BCA = arc ACB / 2

BCA = 146 / 2

BCA = 73 degrees

Therefore, BCA is 73 degrees.

To ensure that triangle ABC exists, we need to consider the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's assume that the lengths of the sides of triangle ABC are represented by the variables a, b, and c.

According to the options provided, the inequality that represents the values of x that ensure triangle ABC exists is:

C. 1 < x < 5

This is because if we substitute x with values within this range, the resulting lengths of the sides will satisfy the triangle inequality theorem.

To prove that the inequality 1 < x < 5 ensures the existence of triangle ABC, we need to show that for any value of x within this range, the lengths of the sides of triangle ABC satisfy the triangle inequality theorem.

The triangle inequality theorem states that for any triangle with side lengths a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side. In other words, for triangle ABC, we have:

a + b > c

b + c > a

c + a > b

Let's consider the inequality 1 < x < 5. This means that x is greater than 1 and less than 5.

To prove that triangle ABC exists for this range of x, we need to show that the lengths of the sides a, b, and c satisfy the triangle inequality theorem.

Let's consider the lengths of the sides in terms of x:

Side a = 2x + 4

Side b = x + 18

Side c = 6x

We will check if the inequalities hold for these side lengths:

a + b > c:

(2x + 4) + (x + 18) > 6x

3x + 22 > 6x

22 > 3x

b + c > a:

(x + 18) + 6x > 2x + 4

7x + 18 > 2x + 4

5x > -14

c + a > b:

6x + (2x + 4) > x + 18

8x + 4 > x + 18

7x > 14

From these inequalities, we can see that for any value of x within the range 1 < x < 5, the side lengths satisfy the triangle inequality theorem. Therefore, triangle ABC exists when 1 < x < 5.

This completes the proof that the inequality 1 < x < 5 ensures the existence of triangle ABC.

Answer: the y-intercept is at four

Step-by-step explanation: 4 is the point which the function cross the y axis on a graph

Answer:

Everything looks right except the first one which is Distributive Property.

Step-by-step explanation:

multiplying out - 3(x - 3) is considered distributive property as you are distributing the -3 to the x and the 3.

-3 * x = -3x and -3 * 3 = -9

With speed of 7.5 mi/hr and 12.5 mi/hr fast he would be going and returning respectively.

To solve this problem, let's denote the speed of Jay's motorcycle on the outbound trip as "x" miles per hour. Since he was able to average 5 mi/hr faster on the return trip, his speed on the return trip would be "x + 5" miles per hour.

We know that the total time for the round trip is 5 hours. The time taken for the outbound trip is the distance divided by the speed, which is 100 / x. The time taken for the return trip is the distance divided by the speed, which is 100 / (x + 5).

According to the problem, the total time for the round trip is 5 hours. Therefore, we can set up the equation:

100 / x + 100 / (x + 5) = 5

To solve this equation, we can multiply through by x(x + 5) to eliminate the denominators:

100(x + 5) + 100x = 5x(x + 5)

Expanding and simplifying the equation, we get:

200x + 500 = 5x^2 + 25x

Bringing all terms to one side and simplifying further, we obtain a quadratic equation:

5x^2 + 25x - 200x - 500 = 0

5x^2 - 175x - 500 = 0

Factoring the equation, we find:

(x - 7.5)(x + 12.5) = 0

So, x = 7.5 or x = -12.5. Since speed cannot be negative, the only valid solution is x = 7.5.

Therefore, Jay traveled at a speed of 7.5 mi/hr on the outbound trip and 7.5 + 5 = 12.5 mi/hr on the return trip.

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