Simplify or analyze the expression |x - 2| + x - 4 for x < 2.

The expression ∣ x − 2∣ + x − 4 simplifies to − 2 for all values of x < 2 . This is because when x < 2 , ∣ x − 2∣ changes the expression to a constant value. Therefore, regardless of the specific value of x in this domain, the result remains − 2 .
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Answered by Anonymous | 2025-07-04

To simplify the expression ∣ x − 2∣ + x − 4 for x < 2 , we must consider the properties of absolute value.
The absolute value function ∣ x − a ∣ is defined as:

x − a if x ≥ a
− ( x − a ) if x < a

Since we are considering the case where x < 2 , we apply the second definition of absolute value:

Replace ∣ x − 2∣ with − ( x − 2 ) since x < 2 .

Simplify the expression:
∣ x − 2∣ + x − 4 = − ( x − 2 ) + x − 4

Distribute the negative sign in − ( x − 2 ) :
= − x + 2 + x − 4

Combine like terms:
= ( − x + x ) + ( 2 − 4 )

Simplify the expression further:
= 0 + ( − 2 )

Therefore, the simplified expression is:
= − 2


When x < 2 , the entire expression simplifies to − 2 . This means for any value of x in this range, the expression evaluates to − 2 .
In conclusion, understanding properties of absolute value helps in simplifying expressions effectively, especially by dealing with piecewise definitions based on the conditions provided.

Answered by MasonWilliamTurner | 2025-07-06