Which of the following expressions is equal to
-5 for some value of x?
A) |x - 6| + 2
B) |x - 2|-6
C) |x+2|+6
D) |x + 6|-2

Explanation:

Recall that for y = a| x-h | + k, the vertex is (h,k).

Since all of the answer choices have a > 0, this means all of the vertexes are the lowest point for each absolute value graph.

Consequently it tells us the lowest possible y value.

Choices A, C and D indicate that getting y = -5 is impossible. For instance, if y = 2 is the smallest, then there's no way to get y = -5. I recommend making a number line.

The only thing left is choice B. If we plug in y = -5 then we get...

y = |x-2| - 6

-5 = |x-2| - 6

-5+6 = |x-2|

|x-2| = 1

x-2 = 1 or x-2 = -1

x = 1+2 or x = -1+2

x = 3 or x = 1

Plugging either of those x values into y = |x-2| - 6 leads to y = -5

You could try solving something like -5 = |x-6| + 2 for x, but you'll find that it has no solutions.

As another alternative, you can graph each answer choice. Then see which one intersects the horizontal line y = -5. You should see that only y = |x-2| - 6 does so.