Examkin

Question

Difficulty: MediumAbsolute Value

If y=−3y = -3, which of the following represents all possible values for xx in the equation −2−2∣x−y∣=−16-2 - 2|x - y| = -16?

  1. A
    1010 and −4-4
  2. B
    11 and −7-7
  3. 44 and −10-10Answer
  4. D
    44

Answer

The possible values for xx are 44 and −10-10.
The correct answer accurately substitutes the negative value, isolates the absolute value term to yield ∣x+3∣=7|x + 3| = 7, and evaluates both the positive and negative branches to find the two valid solutions for xx.

Step-by-Step Solution

1
Substitute the value y=−3y = -3 into the original equation.
−2−2∣x−(−3)∣=−16⇒−2−2∣x+3∣=−16-2 - 2|x - (-3)| = -16 \Rightarrow -2 - 2|x + 3| = -16
To reduce the equation to a single variable so it can be solved.
2
Begin isolating the absolute value expression by adding 22 to both sides.
−2∣x+3∣=−16+2⇒−2∣x+3∣=−14-2|x + 3| = -16 + 2 \Rightarrow -2|x + 3| = -14
To undo the constant subtraction outside the absolute value, following reverse order of operations.
3
Divide both sides of the equation by −2-2.
∣x+3∣=7|x + 3| = 7
To completely isolate the absolute value expression.
4
Set up two separate equations for the positive and negative pathways.
x+3=7x + 3 = 7 and x+3=−7x + 3 = -7
The expression inside the absolute value can evaluate to either 77 or −7-7 and still have an absolute distance of 77 from zero.
5
Solve both resulting linear equations for xx.
x=4x = 4 and x=−10x = -10
To find all possible values that satisfy the original equation.

Key Concept

Solving linear absolute value equations requires accurately substituting variables, strictly following the order of operations to isolate the absolute value, and creating two distinct pathways (positive and negative) to find all solutions.
Estimated Time:1m 30s
Rate this question

Topics

Absolute Value

More questions from this topic