The temperature, t, in degrees Fahrenheit, in a certain town on a certain spring day satisfies the inequality | t - 24 | is less than or equal to 30. Which of the following temperatures, in degrees Fahrenheit, is NOT in this range?

F. -10
G. -6
H. -5
J. 0
K. 54

| t - 24 | ≤ 30

There are two cases here: positive and negative:

case I: positive
t - 24 ≤ 30
t ≤ 30 + 24
t ≤ 54

case II: negative
-(t - 24) ≤ 30
-t + 24 ≤ 30
24 - 30 ≤ t
-6 ≤ t

We get the intersection of these solution:
-6 ≤ t ≤ 54

Now from the choices, which of them is not included in this set?

hgh

-10

I'm confused

Well, if we think about it, the temperature range is between t - 24 and t + 24. So let's go through each option and see which one doesn't fit!

F. -10: Well, if we subtract 24 from -10, we get -34, and that's within the range. So it's not F.

G. -6: If we subtract 24 from -6, we get -30, and that's within the range. So it's not G.

H. -5: Subtracting 24 from -5 gives us -29, and that's within the range. So it's not H.

J. 0: When we subtract 24 from 0, we get -24, and that's within the range. So it's not J.

K. 54: Subtraction 24 from 54 gives us 30, and that's OUTSIDE the range! So the answer is K, 54.

Looks like poor 54 is too hot to hang out with the other temperatures in this range!

To determine which temperature is NOT in the given range, we need to solve the inequality |t - 24| ≤ 30.

Step 1: Set up the two cases and solve individually:
Case 1: t - 24 ≤ 30
Adding 24 to both sides: t ≤ 54

Case 2: -(t - 24) ≤ 30 (We are multiplying both sides by -1 to switch the inequality sign)
Distributing the negative sign: -t + 24 ≤ 30
Subtracting 24 from both sides: -t ≤ 6
Dividing by -1 (remember to reverse the inequality): t ≥ -6

Step 2: Combine the results from both cases:
From Case 1: t ≤ 54
From Case 2: t ≥ -6

So the valid temperature range is -6 ≤ t ≤ 54.

Now, we can check each given temperature option to see which one does not fall within this range.

F. -10: -6 ≤ -10 ≤ 54 (not in range)
G. -6: -6 ≤ -6 ≤ 54 (in range)
H. -5: -6 ≤ -5 ≤ 54 (in range)
J. 0: -6 ≤ 0 ≤ 54 (in range)
K. 54: -6 ≤ 54 ≤ 54 (in range)

Therefore, the temperature NOT in this range is F. -10 degrees Fahrenheit.