Solve the equation for all values of $x$ . $\newline$ $|x+8|=3 x$ $\newline$ Answer: $x=$

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Grade 8

Evaluate absolute value expressions

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Q. Solve the equation for all values of

x

\newline

|x+8|=3 x

\newline

Answer:

x=

Absolute Value Equation Split: We have the equation $|x+8|=3x$ . The absolute value equation can split into two separate equations, one for the positive case and one for the negative case. $\newline$ For the positive case, we have: $\newline$ $x + 8 = 3x$
Positive Case Solution: To solve for $x$ , we need to get all the $x$ terms on one side. Subtract $x$ from both sides of the equation: $\newline$ $x + 8 - x = 3x - x$ $\newline$ $8 = 2x$
Negative Case Solution: Now, divide both sides by $2$ to solve for $x$ : $\frac{8}{2} = \frac{2x}{2}$ $4 = x$
Positive Case Simplification: For the negative case, we have: $\newline$ $- (x + 8) = 3x$ $\newline$ $-x - 8 = 3x$
Positive Case Solution: To solve for $x$ , we need to get all the $x$ terms on one side. Add $x$ to both sides of the equation: $\newline$ $-x - 8 + x = 3x + x$ $\newline$ $-8 = 4x$
Negative Case Simplification: Now, divide both sides by $4$ to solve for $x$ : $-8 / 4 = 4x / 4$ $-2 = x$
Negative Case Solution: We have found two potential solutions for the equation $|x+8|=3x$ : $x = 4$ and $x = -2$ . However, we must check these solutions to ensure they do not violate the original absolute value equation.
Check $x = 4$ : Check $x = 4$ in the original equation: $\newline$ $|4 + 8| = 3 \times 4$ $\newline$ $|12| = 12$ $\newline$ $12 = 12$ which is true.
Check $x = -2$ : Check $x = -2$ in the original equation: $\newline$ $|-2 + 8| = 3 \times -2$ $\newline$ $|6| = -6$ $\newline$ $6 \neq -6$ which is false. Therefore, $x = -2$ is not a solution.