Line Z passes above points (5, 5) and (25, 15) and below point (20, 15

A sketch helps. (A quick, bare-bones sketch.)
Connect point (5,5) and each of the other two points.
Find the slope of those lines using \(\frac{rise}{run}\)

Slope of Z must fall between the numeric values of the other two lines' slopes.

Line Z passes above one line (with Slope A) and below another (with Slope B)
Line Z can be anywhere in the pink shaded area (blue lines not included)

1) Slope A = Lower limit, NOT inclusive, of Line Z's slope

Find the slope of Line Z as IF it passed through points (5,5) and (25,15)
Call that "Slope A."

Slope A = \(\frac{rise}{run}=\frac{y_2-y_1}{x_2-x_1}\)

Slope A = \(\frac{15-5}{25-5}=\frac{10}{20}=\frac{1}{2}\)

Slope A\(=\frac{1}{2}\)
Slope A (lower, non-inclusive limit of Slope Z) < Slope of Line Z

2) Slope B = Upper limit, NOT inclusive, of Line Z's slope

Upper limit, Slope B, is between points (5,5) and (20,15)
Slope B = \(\frac{15-5}{20-5}=\frac{10}{15}=\frac{2}{3}\)

Slope B\(=\frac{2}{3}\) > Slope of Line Z
Slope A < Slope of Line Z < Slope B

On the diagram, Line Z can fall only in the pink shaded area -
that area is defined by the slope inequality below:

\(\frac{1}{2} = .5 <\) Slope of Line Z\(<\frac{2}{3}=.67\)

The answer that falls between the lower and upper limit of the compound inequality is the answer.

Answer choices, use decimals or fraction* comparison

A. 1/4 = .25
B. 3/8 = .375
C. 5/8 = .625
D. 3/4 = .74
E. 7/8 = .875

At the least, know or calculate that 1/8 = .125
Then 3/8 = (3 * .125) = .375, etc.

Eliminate A (.25) and B (.375). Both are < .5 (Z is greater than .5)
Eliminate D (.75) and E (.875). Both are > than .67 (Z is less than .67)

The only answer in the correct range is C) 5/8 (=.625)
.5 < .625(Z) < .67
That works.

Answer C

*FRACTIONS
Eliminate A and B. Too small: smaller than 1/2 (Slope of Z is greater than 1/2)
A) 1/4. Too small. Z > 1/2 > 2/4Lower limit 1/2 = 2/4
B) 3/8. Too small. Z > 1/2 > 4/8
Eliminate D and E. Too great: greater than 2/3 (Slope of Z is less than 2/3)
D) 3/4 = 9/12. Too great. Z < 2/3 < 8/12 < 9/12
E) 7/8 = 21/24. Too great. Z < 2/3 < 16/24 < 21/24 . . . . By POE, answer is C
C) 5/8? 1/2 = 4/8 < 5/8. That works. 2/3 = 16/24, and 5/8 = 15/24. 15/24 (5/8) < 16/24 (6/8) That works, too. Answer C.
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Slope has to be less than 2/3 but greater than 1/2. Only C fits.

Answer: C

[quote="Bunuel"]Line Z passes above points (5, 5) and (25, 15) and below point (20, 15). Which of the following could be the slope of Line Z?

A. 1/4
B. 3/8
C. 5/8
D. 3/4
E. 7/8


1/2<m<2/3
only one option lies in this range and that is option C

I might be missing something, but don't A, B, and C all satisfy the question?

The line only has to pass above (5,5) but can be much flatter and still be below (20,15) and above (25,15).

For example, y=(1/4)x + 9.5

I think this line still matches the requirements of the question. Maybe adding a second point which z passes below would help narrow it down ~(5,6)?

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