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Difficulty: MediumAbsolute Value

Given x=−4x = -4 and y=3y = 3, what is the exact value of the expression 3∣x−y∣−∣y2−x2∣3|x - y| - |y^2 - x^2|?

Answer: 14

Answer

14
By correctly substituting the variables, following the order of operations, and applying the definition of absolute value, the expression simplifies to 3(7)−73(7) - 7, which evaluates precisely to 1414.

Step-by-Step Solution

1
Substitute the given values for xx and yy into the expression.
3∣−4−3∣−∣(3)2−(−4)2∣3|-4 - 3| - |(3)^2 - (-4)^2|
Variable substitution is the first step in evaluating an algebraic expression.
2
Simplify the expression inside the first absolute value bars.
∣−4−3∣=∣−7∣|-4 - 3| = |-7|
Absolute value bars act as grouping symbols, so operations inside them must be performed first.
3
Evaluate the exponents inside the second absolute value bars.
∣(3)2−(−4)2∣=∣9−16∣|(3)^2 - (-4)^2| = |9 - 16|
Following the order of operations, exponents are evaluated before subtraction.
4
Simplify the expression inside the second absolute value bars.
∣9−16∣=∣−7∣|9 - 16| = |-7|
Complete the subtraction inside the grouping symbols.
5
Apply the absolute value function to both simplified terms.
3(7)−73(7) - 7
The absolute value of any non-zero real number is its positive counterpart: ∣−7∣=7|-7| = 7.
6
Perform the final multiplication and subtraction.
21−7=1421 - 7 = 14
Multiplication is performed before subtraction according to the standard order of operations.

Key Concept

Evaluating algebraic expressions involving absolute value, exponents, and signed numbers.
Estimated Time:1m 30s
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Topics

Absolute Value

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