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Question

Difficulty: Very hardAbsolute Value

If x=−3x = -3, y=−5y = -5, and z=2z = 2, what is the value of the expression −2∣x2−∣y−3z∣∣+4x-2|x^2 - |y - 3z|| + 4x?

  1. A
    -26
  2. -16Answer
  3. C
    -52
  4. D
    -8

Answer

-16
The correct answer is found by strictly adhering to the order of operations and properties of absolute values. First, the inner absolute value |-5 - 3(2)| evaluates to |-11|, which is 11. The expression then simplifies to -2|(-3)² - 11| + 4(-3). Squaring -3 gives 9, so the expression inside the outer absolute value becomes 9 - 11, which is -2. The absolute value of -2 is 2. Multiplying this by the leading coefficient of -2 gives -4, and adding 4(-3), which is -12, results in -16.

Step-by-Step Solution

1
Substitute the given variable values into the expression.
-2|(-3)² - |-5 - 3(2)|| + 4(-3)
To prepare the expression for evaluation by replacing variables with their known numeric values.
2
Evaluate the innermost operations following the order of operations, starting with the inner absolute value.
-5 - 3(2) = -5 - 6 = -11, so |-11| = 11
The order of operations (PEMDAS) requires evaluating multiplication before subtraction, and operations inside the innermost grouping symbols first.
3
Evaluate the exponent and substitute the result of the inner absolute value back into the expression.
-2|9 - 11| - 12
Exponents must be evaluated before subtraction within the remaining absolute value.
4
Evaluate the subtraction inside the remaining absolute value and apply the absolute value function.
|9 - 11| = |-2| = 2
The absolute value of any non-zero integer is its positive counterpart.
5
Multiply by the leading coefficient and add the final term.
-2(2) - 12 = -4 - 12 = -16
Multiplication must be performed before subtraction to find the final value.

Key Concept

Evaluating multi-step algebraic expressions with nested absolute values
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Topics

Absolute Value

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