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

Negative numbers and coordinate plane
asked 2021-01-10

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

Answers (1)


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.

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