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Question # Let a and b be positive numbers such that a < b. State whether the absolute value equation

Negative numbers and coordinate plane
ANSWERED Let a and b be positive numbers such that $$a < b$$. State whether the absolute value equation has no solution, two negative solutions, two positive solutions, or one positive and one negative solution.
$$|x − b| = -a$$
no solution
two negative solutions
two positive solutions
one positive and one negative solution 2021-01-11

Consider that a and b are positive number such that $$a < b$$.
Recall that the absolute difference of two number is a positive number.
So, the expression $$| x – b |$$ is always a positive number for all values of x.
But -a is a negative number since a is a positive number.
Since the left-hand side of the equation $$|x − b| = -a$$ is a positive number and right-hand side of the equation $$|x − b| = -a$$ is a negative number hence the equation $$|x − b| = -a$$ has no solution.