Find the domain and range of function

Answer:

There is one solution. The solution is 2, 18, 19.

Step-by-step explanation:

If you want me to show working tell me in the comments and I'll edit the answer

Answer:

A. (2, 18, -19)

Step-by-step explanation:

To solve:

Z is the most suitable variable to remove first

Add the first equation to the second equation: (this conveniently removes both y and z)

(x+y-z) + (4x-y+z) = 1+9

Simplify

5x = 10

Solve

x = 2

Multiply the second equation by 2 and minus it to the third equation: (Solve for y)

2(4x-y+z) - (x-3y+2z) = 2(9) - (-14)

Simplify

8x-2y+2z-x+3y-2z=18+14

7x+y=32

Substitute using x=2

7(2) + y = 32

y = 32 - 14

y = 18

Now substitute x and y for their respective values into Equation 1

2 + (-18) - z = 1

Simplify

-z = 19

z = -19

So :

x = 2, y = 18 , z = -19

The linear inequality represented by the graph is y < 3x + 2. Option A.

To determine the linear inequality represented by the graph, let's analyze the given information and the slope-intercept form of a linear equation (y = mx + b), where m represents the slope and b represents the y-intercept.

We are given two points on the line: (-3, -7) and (0, 2). Using these points, we can calculate the slope (m) as follows:

m = (y2 - y1) / (x2 - x1)

= (2 - (-7)) / (0 - (-3))

= 9 / 3

= 3

Therefore, the slope of the line is 3.

Next, we can substitute the slope and one of the given points into the slope-intercept form to find the y-intercept (b). Let's use the point (0, 2):

y = mx + b

2 = 3(0) + b

2 = b

So, the y-intercept (b) is 2.

Now we have the equation of the line: y = 3x + 2.

The shaded region is to the left of the line. To express this region as an inequality, we need to find the inequality symbol. Since everything to the left of the line is shaded, we need the inequality to represent values less than the line.

Therefore, the correct inequality is y < 3x + 2.

Hence, the linear inequality represented by the graph is y < 3x + 2. So Option A is correct.

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Note the complete question is

On a coordinate plane, a dashed straight line has a positive slope and goes through (negative 3, negative 7) and (0, 2). Everything to the left of the line is shaded.

Which linear inequality is represented by the graph?

A.) y < 3x + 2

B.) y > 3x + 2

C.) y < 1/3x + 2

D.) y > 1/3x + 2

Answer:

math = 227.98

programming = 42.55

Step-by-step explanation:

We have
3m + 4p = 854.14 -eq(1)

8m + 1p = 1866.39 -eq(2)

rq(2) x 4: 32m + 4p = 7465.56 -eq(3)

eq(3)-eq(1):

32m + 4p = 7465.56

- ( 3m + 4p = 854.14)

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

29m = 6611.42

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

⇒ m = 6611.42/29

m = 227.98

sub in eq(1)

3(227.98) + 4p = 854.14

4p = 854.14 - 683.94

4p = 170.2

p = 170.2/4

p = 42.55

Therefore, the combined volume of the carton of milk and the box of cookies is 232 cubic inches.

To find the combined volume of the carton of milk and the box of cookies, we need to calculate the volume of each object and then add them together.

The volume of an object can be found by multiplying its length, width, and height. Let's calculate the volume for each item:

   Carton of milk:

   Volume = Length × Width × Height

   = 6 inches × 4 inches × 8 inches

   = 192 cubic inches

   Box of cookies:

   Volume = Length × Width × Height

   = 5 inches × 4 inches × 2 inches

   = 40 cubic inches

Now, we can find the combined volume by adding the volumes of the carton of milk and the box of cookies:

Combined Volume = Volume of Carton of milk + Volume of Box of cookies

= 192 cubic inches + 40 cubic inches

= 232 cubic inches

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Geno read 36 pages during the first and second hour, and 1.5 times that, which is 54 pages, during the third hour.

Let's break down the information given:

Geno read 126 pages in 3 hours.

He read the same number of pages each hour for the first 2 hours.

Geno read 1.5 times as many pages during the third hour as he did during the first hour.

Let's solve this:

Let's assume that Geno read x pages during the first hour.

Since he read the same number of pages each hour for the first 2 hours, he also read x pages during the second hour.

During the third hour, Geno read 1.5 times as many pages as he did during the first hour, which is 1.5x pages.

To find the total number of pages he read, we can add up the pages from each hour: x + x + 1.5x = 126.

Combining like terms, we have 3.5x = 126.

Divide both sides of the equation by 3.5 to solve for x: x = 36.

Therefore, Geno read 36 pages during the first and second hour, and 1.5 times that, which is 54 pages, during the third hour.

In summary, Geno read 36 pages during each of the first two hours and 54 pages during the third hour, for a total of 126 pages in 3 hours.

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The statement that is true about the function is "The function is negative for all real values of x where x < -2."

To determine the statement that is true about the function f(x) = (x + 2)(x + 6) based on the given graph, we can analyze the behavior of the graph and identify the regions where the function is positive or negative.

Looking at the graph:

The function intersects the x-axis at x = -6 and x = -2.

The graph is below the x-axis between x = -6 and x = -2, and above the x-axis outside of that interval.

From this information, we can conclude that the function is negative for all real values of x where x < -2. This is because the graph is below the x-axis in that region.

Therefore, the statement that is true about the function is:

"The function is negative for all real values of x where x < -2."

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Answer:

8

Step-by-step explanation:

Substitute the values in the expression, we have:

[tex]\displaystyle{|-1-3|+4}[/tex]

Evaluate:

[tex]\displaystyle{|-4|+4}[/tex]

Any real numbers in the absolute sign will always be evaluated as positive values. Thus:

[tex]\displaystyle{|-4|+4 = 4+4}\\\\\displaystyle{=8}[/tex]

Hence, the answer is 8. A quick note that z-value is not used due to lack of z-term in the expression.

Answer:

x = - 3 , x = [tex]\frac{1}{3}[/tex]

Step-by-step explanation:

to find the x- intercepts let y = 0 , that is

(x + 3)(6x - 2) = 0

equate each factor to zero and solve for x

x + 3 = 0 ( subtract 3 from both sides )

x = - 3

6x - 2 = 0 ( add 2 to both sides )

6x = 2 ( divide both sides by 6 )

x = [tex]\frac{2}{6}[/tex] = [tex]\frac{1}{3}[/tex]

the x- intercepts are then x = - 2 and x = [tex]\frac{1}{3}[/tex]

Answer: seconds

Step-by-step explanation:

The graph in this problem is the graph of y = 2sin(x), not y = x, as it has a amplitude of 2.

The standard definition of the sine function is given as follows:

y = Asin(B(x - C)) + D.

For which the parameters are given as follows:

The function in this problem has an amplitude of 2, with no phase shift, no vertical shift and period of 2π, hence it is defined as follows:

y = 2sin(x)

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IQR, because Sunny Town is symmetric should be used for both sets of data to determine the location with the most consistent temperature?(option a).

1. The question asks for the measure of variability that should be used to determine the location with the most consistent temperature when comparing the data from two different locations.

2. The first option, A, suggests using the Interquartile Range (IQR) because Sunny Town is symmetric. This means that the data in Sunny Town is evenly distributed around the median, indicating consistency in temperatures.

3. The second option, B, proposes using the IQR because Beach Town is skewed. Skewness implies an asymmetrical distribution, which may indicate less consistency in temperatures.

4. The third option, C, suggests using the Range because Sunny Town is skewed. Skewed data in Sunny Town might imply a larger spread and less consistency in temperatures.

5. The fourth option, D, recommends using the Range because Beach Town is symmetric. However, symmetric data indicates consistency, making the Range less suitable as a measure of variability.

6. Considering the explanations for each option, the best choice is A, IQR, because Sunny Town is symmetric. The symmetric distribution suggests that the temperatures in Sunny Town are consistent and evenly distributed around the median.

7. Therefore, the measure of variability that should be used for both sets of data to determine the location with the most consistent temperature is the IQR, as indicated by option A.

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Answer:

27)  34.29 in²

28)  If I get an A, then I studied for my final.

Step-by-step explanation:

To calculate the area of the trapezoid, we need to find its perpendicular height.

As the given diagram shows an isosceles trapezoid (since the non-parallel sides (the legs) are of equal length), we can use Pythagoras Theorem to calculate the perpendicular height.

Identify the right triangle formed by drawing the perpendicular height from the vertex of the bottom base to the top base (this has been done for you in the given diagram).

As the two base angles of an isosceles trapezoid are always congruent, the base of the right triangle is half the difference between the lengths of the parallel bases, which is (8 - 6)/2 = 1 inch.

The hypotenuse of the right triangle is the leg of the trapezoid, which is 5 inches.

Use Pythagoras Theorem to find the perpendicular height (the length of the other leg):

[tex]h^2+1^2=5^2[/tex]

[tex]h^2+1=25[/tex]

      [tex]h^2=24[/tex]

        [tex]h=\sqrt{24}[/tex]

        [tex]h=2\sqrt{6}[/tex]

Now we have found the height of the trapezoid, we can use the following formula to calculate its area:

[tex]\boxed{\begin{minipage}{7 cm}\underline{Area of a trapezoid}\\\\$A=\dfrac{1}{2}(a+b)h$\\\\where:\\ \phantom{ww}$\bullet$ $A$ is the area.\\ \phantom{ww}$\bullet$ $a$ and $b$ are the parallel sides (bases).\\\phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

The values to substitute into the area formula are:

Substituting these values into the formula we get:

[tex]A=\dfrac{1}{2}(8+6) \cdot 2\sqrt{6}[/tex]

[tex]A=\dfrac{1}{2}(14) \cdot 2\sqrt{6}[/tex]

[tex]A=7\cdot 2\sqrt{6}[/tex]

[tex]A=14\sqrt{6}[/tex]

[tex]A=34.29\; \sf in^2\;(nearest\;hundredth)[/tex]

Therefore, the area of the isosceles trapezoid is 34.29 in², rounded to the nearest hundredth.

[tex]\hrulefill[/tex]

Given conditional statement:

The hypothesis is "I studied for my final", and the conclusion is "I will get an A".

The converse of a conditional statement involves switching the hypothesis ("if" part) and the conclusion ("then" part) of the original statement.

Therefore, the converse of the statement would be:

To find the local and absolute maximum and minimum points of the function f(x) = (3/5)x^5 - 9x^3 + 2 on the closed interval [-4,5], we need to follow these steps:

a) Find all critical numbers (x-coordinates only):

To find the critical numbers, we need to identify where the derivative of the function is zero or undefined. Let's find the derivative of f(x) first:

f'(x) = 3x^4 - 27x^2

Now, set the derivative equal to zero and solve for x:

3x^4 - 27x^2 = 0

Factoring out a common factor of 3x^2, we get:

3x^2(x^2 - 9) = 0

This equation is satisfied when either 3x^2 = 0 or x^2 - 9 = 0.

For 3x^2 = 0, we have x = 0.

For x^2 - 9 = 0, we have x = -3 and x = 3.

Therefore, the critical numbers (x-coordinates) are 0, -3, and 3.

b) Find the intervals on which the graph is increasing (mark critical numbers):

To determine the intervals of increasing, we need to analyze the sign of the derivative on each side of the critical numbers. We create a sign chart for f'(x):

Interval (-∞, -3): Choose a test point x < -3, e.g., x = -4

f'(-4) = 3(-4)^4 - 27(-4)^2 = 768 > 0

The derivative is positive, indicating the graph is increasing.

Interval (-3, 0): Choose a test point x between -3 and 0, e.g., x = -1

f'(-1) = 3(-1)^4 - 27(-1)^2 = -24 < 0

The derivative is negative, indicating the graph is decreasing.

Interval (0, 3): Choose a test point x between 0 and 3, e.g., x = 1

f'(1) = 3(1)^4 - 27(1)^2 = -24 < 0

The derivative is negative, indicating the graph is decreasing.

Interval (3, ∞): Choose a test point x > 3, e.g., x = 4

f'(4) = 3(4)^4 - 27(4)^2 = 768 > 0

The derivative is positive, indicating the graph is increasing.

Therefore, the graph is increasing on the intervals (-∞, -3) and (3, ∞).

c) Find the intervals on which the graph is decreasing (mark critical numbers):

From the analysis above, we can see that the graph is decreasing on the intervals (-3, 0) and (0, 3).

d) Find all local maximum points:

To find the local maximum points, we need to examine the points where the graph changes from increasing to decreasing. By observing the sign changes in the derivative, we can identify potential local maximum points.

From our analysis in part b, we can see that the graph changes from increasing to decreasing at x = -3 and x = 0. Therefore, these are the local maximum points.

e) Find all local minimum points:

To find the local minimum points, we need to examine the points where the graph changes from decreasing to increasing. By observing the sign changes in the derivative, we can identify potential local minimum points.

From our analysis in part c, we can see that the graph changes.

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Answer:

x = 5

Step-by-step explanation:

We can solve for x in this proportion, or an equation of ratios, by multiplying both sides by 20.

[tex]{/}\!\!\!\!\!{20}\cdot \dfrac{5}{{/}\!\!\!\!\!20}=\dfrac{x}{{/}\!\!\!\!\!20} \cdot {/}\!\!\!\!\!20[/tex]

We can see that the 20s cancel in the numerator and denominator in both sides, and the equation is solved for x:

[tex]x = 5[/tex]

Answer:

m∠BEC + m∠CED = m∠BED; Angle Addition Postulate

Step-by-step explanation:

You need to show that <BED is made up of angles BEC and CED by the Angle Addition Postulate.

m∠BEC + m∠CED = m∠BED; Angle Addition Postulate

The surface area of the cone is: 213.66 cm²

The volume of a cone is: 7.33 cm³

The formula for the surface area of a cone is:

T.S.A = πrl + πr²

where:

r is radius

l is slant length

From the diagram and using Pythagoras theorem,we have:

l = √(7² + 5²)

l = √74

Thus:

TSA =  (π * 5 * √74) + (π * 5²)

TSA = 213.66 cm²

Formula for the volume of a cone is:

V = ¹/₃πr²

V = ¹/₃π * 7

V = 7.33 cm³

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The equation y = -6x + 2 (option c) is parallel to the graph of y = -6x + 3.

The slope-intercept form is expressed as;

y = mx + b

Where m is slope and b is the y-intercept.

Given the equation of the graph in the question:

y = -6x + 3

To determine which of the given options:

a) y = (1/6)x + 3

b) y = -(1/6) + 3

c) y = -6x + 2

d) y = 3x - 6

is parallel to the graph of y = -6x + 3, we need to compare their slopes.

The given equation of the graph is y = -6x + 3:

Slope of the graph is -6.

Now, lets check each option:

a) y = (1/6)x + 3

This equation has a slope of 1/6, which is not equal to -6.

Therefore, it is not parallel to y = -6x + 3.

b) y = -(1/6) + 3

This equation also has a slope of 1/6 (the negative sign doesn't affect the slope), it is not parallel to y = -6x + 3.

c) y = -6x + 2

This equation has a slope of -6, which is the same as the slope of y = -6x + 3. Therefore, it is parallel to the given graph.

d) y = 3x - 6

This equation has a slope of 3, which is not equal to -6. Thus, it is not parallel to y = -6x + 3.

Therefore option C) y = -6x + 2 is the correct answer.

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The system has exactly one solution. The solution is (13, 8)

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

Country A: -x + 20y = 147

Country B: -x + 10y = 67

Country C: y = 8

So, we have

-x + 20y = 147

-x + 10y = 67

y = 8

Substitute 8 for y in the first and second equations

So, we have

-x + 20 * 8 = 147

-x + 10 * 8 = 67

Evaluate the products

-x + 160 = 147

-x + 80 = 67

So, we have

x = 160 - 147

x = 80 - 67

Evaluate

x = 13

x = 13

Hence, the solution to the system of equations is (13, 8)

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Answer:

Step-by-step explanation:Angle EDH=Angle FDH, so A must be correct.

Also, we don't have more information to prove B, C, D is right

Answer:

A.

Step-by-step explanation:

An angle bisector is a line, ray, or segment that divides an angle into two equal parts. It divides the angle into two congruent or equal angles. The angle bisector originates from the vertex of the angle and extends towards the interior of the angle. It essentially cuts the angle into two smaller angles of equal measure.

Using row operations to write the augmented matrix, the value of x, y and z are 2, -12 and 11

To solve the system using row operations, we'll write the augmented matrix and perform row operations to transform it into row-echelon form. Here are the steps:

1. Write the augmented matrix for the system of equations:

  [1  1  -1  |  1]

  [4  -1  1  |  9]

  [1  -3  2  |  -14]

2. Perform row operations to transform the matrix into row-echelon form:

  R2 = R2 - 4R1

  R3 = R3 - R1

  [1  1   -1  |  1]

  [0  -5  5   |  5]

  [0  -4  3   |  -15]

3. Perform row operations to further transform the matrix into row-echelon form:

  R2 = -R2/5

  R3 = -4R2 + R3

  [1  1   -1  |  1]

  [0  1   -1  |  -1]

  [0  0   -1  |  -11]

4. Perform row operations to obtain a diagonal of 1s from left to right:

  R1 = R1 + R3

  R2 = R2 + R3

  [1  1   0  |  -10]

  [0  1   0  |  -12]

  [0  0   -1 |  -11]

5. Perform row operations to transform the matrix into reduced row-echelon form:

  R3 = -R3

  [1  1   0  |  -10]

  [0  1   0  |  -12]

  [0  0   1  |  11]

The resulting matrix corresponds to the system of equations:

x + y = -10

y = -12

z = 11

Therefore, the solution to the given system of equations is x = -10 - y, y = -12, and z = 11.

So, the solution is x = -10 - (-12) = 2, y = -12, and z = 11.

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Answer:

Step-by-step explanation:

The given augmented matrix is already in reduced row echelon form. We can interpret it as a system of equations as follows:

1x + 0y + 0z + (4/5)w = 0

0x + 1y + 0z + 5w = 0

0x + 0y + 1z - 4w = 0

0x + 0y + 0z + w = -4

From the last row, we can see that w = -4. Substituting this value back into the previous rows, we get:

1x + 0y + 0z + (4/5)(-4) = 0

0x + 1y + 0z + 5(-4) = 0

0x + 0y + 1z - 4(-4) = 0

Simplifying these equations, we have:

x - (16/5) = 0

y - 20 = 0

z + 16 = 0

From the second equation, y = 20. From the third equation, z = -16. Substituting these values into the first equation, we get x - (16/5) = 0, which implies x = 16/5.

Therefore, the solution of the system is (16/5, 20, -16, -4).

So, the correct choice is:

A. The system has exactly one solution. The solution is

(16/5, 20, -16, -4)

Answer:

Step-by-step explanation:

To calculate the amount Aditi needs to pay, we need to apply the discounts step by step based on the given offer.

Step 1: 20% discount on all products

The marked price of the goods is Rs 2400. Applying a 20% discount means she will get a reduction of 20% of the marked price.

20% of Rs 2400 = (20/100) * Rs 2400 = Rs 480

After the first discount, the new bill amount is Rs 2400 - Rs 480 = Rs 1920.

Step 2: Additional 20% discount on the bill amount if it exceeds Rs 1000

The new bill amount after the first discount is Rs 1920. If this amount exceeds Rs 1000, Aditi will get a further discount of 20% on this bill amount.

Since Rs 1920 is greater than Rs 1000, we can apply a 20% discount to it.

20% of Rs 1920 = (20/100) * Rs 1920 = Rs 384

The final amount Aditi needs to pay after both discounts is Rs 1920 - Rs 384 = Rs 1536.

Therefore, Aditi needs to pay Rs 1536.

Based on the data and the one-sample t-test, at the 0.05 level of significance, there is sufficient evidence to conclude that the average bear sales per store at Wal-Mart is significantly higher than $500

.

To determine if there is evidence that the average bear sales per store at Wal-Mart is more than $500 at the 0.05 level of significance, we can conduct a one-sample t-test. Let's go through the steps:

State the null and alternative hypotheses:

Null hypothesis (H₀): The average bear sales per store is equal to or less than $500.

Alternative hypothesis (H₁): The average bear sales per store is greater than $500.

Set the significance level (α):

In this case, the significance level is given as 0.05 or 5%.

Collect and analyze the data:

The weekly sales of bears in 10 stores are as follows:

8, 11, 0, 4, 7, 8, 10, 583

Calculate the test statistic:

To calculate the test statistic, we need to compute the sample mean, sample standard deviation, and the standard error of the mean.

Sample mean ([tex]\bar X[/tex]):

[tex]\bar X[/tex] = (8 + 11 + 0 + 4 + 7 + 8 + 10 + 583) / 8

[tex]\bar X[/tex] ≈ 76.375

Sample standard deviation (s):

s = √[Σ(x - [tex]\bar X[/tex])² / (n - 1)]

s ≈ 190.687

Standard error of the mean (SE):

SE = s / √n

SE ≈ 60.174

Now, we can calculate the t-value:

t = ([tex]\bar X[/tex] - μ₀) / SE

Where μ₀ is the hypothesized population mean ($500).

t = (76.375 - 500) / 60.174

t ≈ -7.758

Determine the critical value:

Since we are conducting a one-tailed test and the alternative hypothesis is that the average bear sales per store is greater than $500, we need to find the critical value for a one-tailed t-test with 8 degrees of freedom at a 0.05 level of significance. Looking up the critical value in the t-distribution table, we find it to be approximately 1.860.

Compare the test statistic with the critical value:

Since -7.758 is less than -1.860, we have enough evidence to reject the null hypothesis.

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Rose's age should be a whole number, we can round 7.8 to the nearest whole number, which is 8.

Let's assume the number of primroses in the vase is p, and the number of celandines is c.

Each primrose has 5 petals, so the total number of primrose petals is 5p.

Each celandine has 8 petals, so the total number of celandine petals is 8c.

According to the given information, the total number of petals is 39. Therefore, we can set up the equation:

5p + 8c = 39 (Equation 1)

Aunt Lily mentions that the total number of flowers is exactly Rose's age. Since Rose's age is not provided, we'll represent it with the variable r.

The total number of flowers is p + c, which is also equal to Rose's age (r). Therefore, we have another equation:

p + c = r (Equation 2)

We need to find the value of r (Rose's age). To do that, we'll solve the system of equations by eliminating one variable.

Multiplying Equation 2 by 5, we get:

5p + 5c = 5r (Equation 3)

Now we can subtract Equation 1 from Equation 3 to eliminate the p term:

(5p + 5c) - (5p + 8c) = 5r - 39

This simplifies to:

-3c = 5r - 39

Now, let's rearrange Equation 2 to solve for p:

p = r - c (Equation 4)

Substituting Equation 4 into the simplified form of Equation 3, we have:

-3c = 5r - 39

Substituting r - c for p, we get:

-3c = 5(r - c) - 39

Expanding, we have:

-3c = 5r - 5c - 39

Rearranging the terms, we get:

2c = 5r - 39

Now we have a system of two equations:

-3c = 5r - 39 (Equation 5)

2c = 5r - 39 (Equation 6)

To solve this system, we can eliminate one variable by multiplying Equation 5 by 2 and Equation 6 by 3:

-6c = 10r - 78 (Equation 7)

6c = 15r - 117 (Equation 8)

Now, let's add Equation 7 and Equation 8 to eliminate c:

-6c + 6c = 10r + 15r - 78 - 117

This simplifies to:

25r = 195

Dividing both sides by 25, we get:

r = 7.8

Since Rose's age should be a whole number, we can round 7.8 to the nearest whole number, which is 8.

Therefore, Rose is 8 years old.

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Answer:

[tex]x = 5\sqrt3[/tex]

Step-by-step explanation:

We can solve for the side length x in this 30-60-90 triangle by using the ratio of side lengths for that specific type of right triangle:

In this triangle, we can identify the smallest side (corresponding to 1 in the ratio) as 5. This means we can solve for x by multiplying 5 by [tex]\sqrt3[/tex]. Thus:

[tex]\boxed{x = 5\sqrt3}[/tex]

Answer:

Its b i bealive

Step-by-step explanation:

Answer:

16cm

Step-by-step explanation:

perimeter for C is 44cm

perimeter for A and B 60cm

60cm-44cm=16cm

Hope this helps

Each person runs 1/3 of a mile when Jeff catches up to Roger.

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Explanation

x = number of minutes that Jeff runs

x+1 = number of minutes Roger runs

Roger has the head start of 1 minute, so he has been running for 1 minute longer compared to Jeff.

Roger runs 1 mile in 9 minutes. His unit rate is 1/9 of a mile per minute.

Jeff's unit rate is 1/6 of a mile per minute.

Let's set up a table with what we have so far

[tex]\begin{array}{|c|c|c|c|} \cline{1-4} & \text{Distance} & \text{rate} & \text{time}\\\cline{1-4}\text{Jeff} & d & 1/6 & \text{x}\\\cline{1-4}\text{Roger} & d & 1/9 & \text{x}+1\\\cline{1-4}\end{array}[/tex]

The distance equation for Jeff is d = (1/6)x

The distance equation for Roger is d = (1/9)(x+1)

note: distance = rate*time

Both runners travel the same distance when Jeff catches up to Roger, so both "d"s are the same value at this specific moment. Set the right hand sides equal to each other and solve for x.

(1/6)x = (1/9)(x+1)

18*(1/6)x = 18*(1/9)(x+1)

3x = 2(x+1)

3x = 2x+2

3x-2x = 2

x = 2

Jeff runs for 2 minutes when he catches up to Roger.

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Check:

Jeff runs for 2 minutes, at 1/6 of a mile per minute, so he runs 2*(1/6) = 2/6 = 1/3 of a mile.

Roger runs for 2+1 = 3 minutes (remember he gets the head start) at 1/9 of a mile per minute, so he has run 3*(1/9) = 3/9 = 1/3 of a mile as well.

Both men have run the same distance which confirms Jeff catches up to Roger at this point. The answer is confirmed.