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Write the absolute value inequality in the form |x-b| < c or |x-b| > c that has the solution set x < -7 or x > 5.
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Answer. b = -1, c = 6.
Plots y = |x - (-1)| (red) and y = 6 (green)
Solution
The problem asks to find the values "b" and "c" in a way that
the solutions of the equation |x - b| = c are the sets {x | x < -7} and {x | x > 5}.
It means that "b" is the center of the segment [-7,5].
This segment has the length 5 - (-7) = 12. Hence, the half of this length is = 6.
Therefore, the center of the segment is -7 + 6 = -1.
Thus the value of "b" is found: it is b = -1.
Then the value of "c" is the distance from (-1) to 5, i.e. c = 6 (same as the distance from (-1) to -7).
Solved.