The high temperature outside one day is 100 degrees. The high temperature decreases at a constant rate of 4 degrees per week. How many weeks could have passed if the high temperature is now above 40 degrees?

Question

Write an inequality to represent the situation. Use x to represent the number of weeks.

100−4x>40
Nice!
You just wrote an inequality to represent the situation. Now solve the associated equation, 100 - 4x = 40, to find the boundary for the inequality. Show your work or solve on paper.

100 - 4x > 40
100−4x
=
40

x
=

15
Good work!
You found that x = 15 is the solution to the associated equation and the boundary of the inequality. Does the boundary, x = 15, make the inequality true or false?
x= 14 x=15 (boundary) x= 16
100 - 4x > 40 ?


True
False
?

Answer 1

Let's evaluate the inequality \(100 - 4x > 40\) at the boundary \(x = 15\).

Step 1: Substitute \(x = 15\) into the inequality

\[ 100 - 4(15) > 40 \]

Step 2: Calculate

\[ 100 - 60 > 40 \]

\[ 40 > 40 \]

Conclusion

Since \(40\) is not greater than \(40\), the statement is false.

Thus, at \(x = 15\):

• \(100 - 4x > 40\) is false.

Now to check the points around the boundary:

• For \(x = 14\):

  • \(100 - 4(14) = 100 - 56 = 44 > 40\) (True)

• For \(x = 16\):

  • \(100 - 4(16) = 100 - 64 = 36 > 40\) (False)

So, for the inequality \(100 - 4x > 40\):

• \(x = 14\) (True)
• \(x = 15\) (False - this is the boundary)
• \(x = 16\) (False)

The solution to the inequality \(100 - 4x > 40\) is: \[ x < 15 \]

Therefore, the number of weeks that could have passed if the high temperature is now above 40 degrees is \(x\) can be any integer from \(0\) to \(14\).