2x + 1 = 2x – 1 how many solutions?
The Correct Answer and Explanation is:
To solve the equation ( 2x + 1 = 2x – 1 ), let’s try to isolate ( x ) by simplifying both sides.
- Subtract ( 2x ) from both sides: [
2x + 1 – 2x = 2x – 1 – 2x
] This simplifies to: [
1 = -1
] - We end up with the statement ( 1 = -1 ), which is a contradiction. This statement is false and has no possible solutions.
Explanation
An equation with no solutions is called an inconsistent equation. In this case, after simplifying the equation, we encountered a contradictory statement, ( 1 = -1 ). Since no value of ( x ) will ever make ( 1 = -1 ) true, there is no possible solution for ( x ).
When solving equations, if the variable terms cancel each other out, you may be left with one of two cases:
- True statement (identity): If the remaining equation is always true (like ( 0 = 0 )), the original equation has infinitely many solutions because it holds for all values of the variable.
- False statement (contradiction): If the remaining equation is always false (like ( 1 = -1 )), the original equation has no solutions because no value of the variable will satisfy it.
In this case, ( 2x + 1 = 2x – 1 ) simplifies to a contradiction. Hence, the equation has no solutions.
In summary, this equation is inconsistent and has no solutions because, after isolating the variable terms, we are left with a false statement that cannot be satisfied by any value of ( x ).