Question

How do you know if an equation is correct for all values of x without solving it?

Answer 1

To determine if an equation is correct for all values of x without solving it, one can use mathematical properties such as identity, contradiction, or conditional statements based on the structure of the equation.

Step-by-step explanation:

Equations can be validated without solving them by analyzing their structure and properties. One approach is to check if the equation is an identity, meaning it holds true for all values of x. An identity is an equation that remains true regardless of the values plugged into it. For example,
`x^2 - y^2 = (x + y)(x - y)` is an identity since it holds for all x and y. To verify, expand the right side and simplify to confirm equality.

Another method is to assess if the equation is a contradiction, which means it is never true for any x. For instance, x + 1 = x is a contradiction as there is no value of x that satisfies this equation. To demonstrate, subtract x from both sides to reveal 1 = 0, highlighting its contradictory nature.

Lastly, an equation can be conditionally true for specific values of x. For instance, x^2 = 4 is true only when x is either 2 or -2. To confirm, square both sides and solve for x, yielding the specific values. In conclusion, by understanding the nature of identities, contradictions, and conditional statements, one can ascertain the validity of an equation without explicitly solving for x.