Explain why, when solving absolute value equations, you must set up both the positive and the negative equations.
The Correct Answer and Explanation is:
When solving absolute value equations, you must set up both the positive and negative equations because the definition of absolute value involves the distance from zero, which can be positive or negative. Absolute value refers to the magnitude of a number, regardless of its sign. For example, the absolute value of both 5 and -5 is 5, as both numbers are 5 units away from zero on a number line.
Consider the equation ( |x| = a ), where ( a ) is a positive number. This equation states that the distance of ( x ) from zero is ( a ), but it doesn’t specify whether ( x ) is to the right (positive) or left (negative) of zero. Therefore, ( x ) could either be ( a ) or ( -a ). This is why you set up two equations:
- ( x = a ) for the positive scenario
- ( x = -a ) for the negative scenario
This dual approach ensures that both possibilities are considered. For example, if you solve ( |x| = 5 ), you would have:
- ( x = 5 ) (positive value)
- ( x = -5 ) (negative value)
Both values satisfy the original equation because the absolute value of both 5 and -5 is 5.
In general, the procedure of setting up two equations stems from the fact that absolute value measures magnitude only, ignoring whether the original number was positive or negative. Therefore, when solving absolute value equations, you must account for both possibilities to fully solve the equation. Failing to do so would result in missing solutions.
For a more complex equation like ( |x + 3| = 7 ), you would set up:
- ( x + 3 = 7 )
- ( x + 3 = -7 )
This ensures that you cover both possible scenarios where the absolute value condition holds.