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Solve the equation for all values of x. |x-4|=3x Answer: x=

Solve the equation for all values of x x .\newline∣x−4∣=3x |x-4|=3 x \newlineAnswer: x= x=

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Evaluate absolute value expressionsright chevron icon

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Q. Solve the equation for all values of x x .\newline∣x−4∣=3x |x-4|=3 x \newlineAnswer: x= x=
  1. Split Absolute Value: We have the equation ∣x−4∣=3x|x - 4| = 3x. The absolute value equation can be split into two separate equations, one for the positive case and one for the negative case.\newlineFor the positive case, we have x−4=3xx - 4 = 3x.
  2. Solve Positive Case: Solve the positive case equation x−4=3xx - 4 = 3x. Subtract xx from both sides to get x−4−x=3x−xx - 4 - x = 3x - x. This simplifies to −4=2x-4 = 2x.
  3. Negative Case Simplification: Divide both sides of −4=2x-4 = 2x by 22 to solve for xx. \newline−4/2=2x/2-4 / 2 = 2x / 2\newlineThis gives us x=−2x = -2.
  4. Solve Negative Case: Now, we need to solve the negative case of the absolute value equation. For the negative case, we have −(x−4)=3x- (x - 4) = 3x.\newlineThis simplifies to −x+4=3x-x + 4 = 3x.
  5. Check Positive Solution: Solve the negative case equation −x+4=3x-x + 4 = 3x. Add xx to both sides to get −x+x+4=3x+x-x + x + 4 = 3x + x. This simplifies to 4=4x4 = 4x.
  6. Check Negative Solution: Divide both sides of 4=4x4 = 4x by 44 to solve for xx. \newline44=4x4\frac{4}{4} = \frac{4x}{4}\newlineThis gives us x=1x = 1.
  7. Final Solution: Check both solutions in the original equation to ensure they do not result in a negative number being equal to a positive number.\newlineFirst, check x=−2x = -2 in ∣x−4∣=3x|x - 4| = 3x.\newline∣−2−4∣=3(−2)|-2 - 4| = 3(-2)\newline∣−6∣=−6| -6 | = -6\newline6=−66 = -6\newlineThis is not true, so x=−2x = -2 is not a solution.
  8. Final Solution: Check both solutions in the original equation to ensure they do not result in a negative number being equal to a positive number.\newlineFirst, check x=−2x = -2 in ∣x−4∣=3x|x - 4| = 3x.\newline∣−2−4∣=3(−2)|-2 - 4| = 3(-2)\newline∣−6∣=−6| -6 | = -6\newline6=−66 = -6\newlineThis is not true, so x=−2x = -2 is not a solution.Now, check x=1x = 1 in ∣x−4∣=3x|x - 4| = 3x.\newline∣1−4∣=3(1)|1 - 4| = 3(1)\newline∣−3∣=3| -3 | = 3\newline∣x−4∣=3x|x - 4| = 3x00\newlineThis is true, so x=1x = 1 is a solution.

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