Near sunset, Emek and Tara reached an abandoned mountain research station and tried to restart its emergency generator. The control panel displayed its operating efficiency as (x^2 - 25)/(x^2 - 2x - 15), where x represented an adjustable fuel-flow setting. Tara factored the numerator as (x - 5)(x + 5) and the denominator as (x - 5)(x + 3). This showed that the expression simplified to (x + 5)/(x + 3), but Emek pointed out that the original denominator was zero at x = 5 and x = -3, so both values remained excluded. The generator required an efficiency value of at least 1.2. They tested the available positive settings 7 and 12. At x = 7, the simplified expression gave 12/10 = 1.2; at x = 12, it gave 17/15, about 1.13. Since 7 met the requirement and was not excluded, they selected that setting. The generator started, and lights flickered on throughout the station before darkness reached the mountain.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4
Hint
Restrictions come from values that make the denominator of the original expression equal to 0.
Why must x = 5 remain excluded even after (x^2 - 25)/(x^2 - 2x - 15) is simplified to (x + 5)/(x + 3)?
Because substituting 5 into the simplified expression gives a negative value.Because x = 5 made the denominator of the original expression equal to 0.Because x = 5 makes the numerator of the original expression equal to 0.Because factors that cancel can never contain positive values.
Correct. Simplifying does not remove restrictions from the original denominator.
Not quite. Check which values make the original denominator equal to 0 before any factors are canceled.
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Hint
Substitute 9 into the simplified expression first, simplify the fraction, and then multiply by 6.
Suppose the generator offers another positive setting x = 9. Using the simplified efficiency expression (x + 5)/(x + 3), multiply the efficiency at x = 9 by 6. What integer do you obtain?
Correct. The efficiency is 7/6, and multiplying by 6 gives 7.
Check the substitution into both the numerator and denominator, then simplify before multiplying.
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Hint
Compare each value with 1.2. You can write 1.2 as 6/5 if that helps.
The generator requires an efficiency of at least 1.2. Which statement correctly compares the available settings x = 7 and x = 12?
Both settings qualify because both efficiency values are greater than 1.Only x = 12 qualifies because 17/15 is greater than 12/10.Only x = 7 qualifies because 12/10 = 1.2, while 17/15 is less than 1.2.Neither setting qualifies because both efficiency values are less than 1.2.
Correct. The setting x = 7 meets the minimum exactly, while x = 12 falls below it.
Not quite. Compare each efficiency value directly with the required minimum of 1.2.
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Dağ İstasyonunun Kilidini Açmak
TR
ARC_G11_OPS_02_STORY_02
Dağ İstasyonunun Kilidini Açmak
Unlocking the Mountain Station
EN
ARC_G11_OPS_02_STORY_02
Unlocking the Mountain Station