VJesus12 wrote: ↑Thu Jun 04, 2020 7:20 am
If \(y = |x – 1|\) and \(y = 3x + 3,\) then \(x\) must be between which of the following values?
(A) 2 and 3
(B) 1 and 2
(C) 0 and 1
(D) –1 and 0
(E) –2 and –1
[spoiler]OA=D[/spoiler]
Solution:
We can set |x - 1| = 3x + 3 and solve for x. However, we can also rewrite the equation as |x - 1| - 3x - 3 = 0 and let f(x) = |x - 1| - 3x - 3. The argument now is: since f(x) is a continuous function and the values of f(a) and f(b) have opposite signs (i.e., one is positive and the other is negative), then there must be a value of x between a and b such that f(x) = 0.
With this in mind, let’s check the answer choices and let’s start with C first. If it works, we’ve found our answer. If it doesn’t, we can still see that we should move forward or backward.
f(0) = |0 - 1| - 3(0) - 3 = 1 - 0 - 3 = -2
f(1) = |1 - 1| - 3(1) - 3 = 0 - 3 - 3 = -6
We see both values are negative and there are no opposite signs. C is not the correct answer. Now let’s move on to D (notice that since we want a positive sign, the term -3x would be positive when x = -1).
f(-1) = |-1 - 1| - 3(-1) - 3 = 2 + 3 - 3 = 2
We see that f(0) = -2 and f(-1) = 2 have opposite signs in their values, there must be a value of x between 0 and -1 such that f(x) = 0 or |x - 1| = 3x + 3.
Alternate Solution:
Since y = |x - 1|, it must be true that y = x - 1 or y = 1 - x.
If y = x - 1, then solving x - 1 = 3x + 3; we obtain 2x = -4, which is equivalent to x = -2. If x = -2; on the other hand, |x - 1| will not equal x - 1. Indeed, x = -2 produce different values for y when we substitute it in the equalities y = |x - 1| and y = 3x + 3.
If y = 1 - x, then solving 1 - x = 3x + 3; we obtain 4x = -2, which is equivalent to x = -1/2. Substituting x = -1/2, we get |-1/2 - 1| = 3(-1/2) + 3 = 3/2. Thus, the only possible value for x is -1/2. We see that x is between -1 and 0.
Answer: D
