SOLUTION: Write the absolute value inequality in the form |x&#8722;b| < c or |x&#8722;b|>c that has the solution set x<&#8722;7 or x>5.

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Question 1073200: 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.
Answer by ikleyn(52798) About Me  (Show Source):
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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 12%2F2 = 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.