Solve the equation for all values of x. |3x+2|-2=x Answer: x=

Consider Non-Negative Case: We have the equation |3x + 2| - 2 = x. To solve for x, we need to consider two cases because of the absolute value: one where 3x + 2 is non-negative, and one where 3x + 2 is negative.Simplify Non-Negative Case: First, let's consider the case where 3x + 2 is non-negative. In this case, the absolute value function does not change the sign of 3x + 2, so we can write the equation without the absolute value bars: 3x + 2 - 2 = xIsolate x in Non-Negative Case: Simplify the equation by subtracting 2 from both sides: 3x = xConsider Negative Case: Subtract x from both sides to solve for x: 3x - x = 0 2x = 0Distribute Negative Case: Divide both sides by 2 to isolate x: x = 0 This is the solution for the case where 3x + 2 is non-negative.Combine Like Terms Negative Case: Now, let's consider the case where 3x + 2 is negative. In this case, the absolute value function changes the sign of 3x + 2, so we can write the equation as: -(3x + 2) - 2 = xSolve for x Negative Case: Distribute the negative sign inside the parentheses: -3x - 2 - 2 = xIsolate x Negative Case: Combine like terms by adding 2 to both sides: -3x - 4 = xAdd 3x to both sides to solve for x: -4 = 4xDivide both sides by 4 to isolate x: x = -4 / 4 x = -1 This is the solution for the case where 3x + 2 is negative.

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Grade 8

Evaluate absolute value expressions

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Q. Solve the equation for all values of

x

\newline

|3 x+2|-2=x

\newline

Answer:

x=

Consider Non-Negative Case: We have the equation $|3x + 2| - 2 = x$ . To solve for $x$ , we need to consider two cases because of the absolute value: one where $3x + 2$ is non-negative, and one where $3x + 2$ is negative.
Simplify Non-Negative Case: First, let's consider the case where $3x + 2$ is non-negative. In this case, the absolute value function does not change the sign of $3x + 2$ , so we can write the equation without the absolute value bars: $\newline$ $3x + 2 - 2 = x$
Isolate $x$ in Non-Negative Case: Simplify the equation by subtracting $2$ from both sides: $3x = x$
Consider Negative Case: Subtract $x$ from both sides to solve for $x$ : $\newline$ $3x - x = 0$ $\newline$ $2x = 0$
Distribute Negative Case: Divide both sides by $2$ to isolate $x$ : $x = 0$ This is the solution for the case where $3x + 2$ is non-negative.
Combine Like Terms Negative Case: Now, let's consider the case where $3x + 2$ is negative. In this case, the absolute value function changes the sign of $3x + 2$ , so we can write the equation as: $\newline$ $-(3x + 2) - 2 = x$
Solve for x Negative Case: Distribute the negative sign inside the parentheses: $-3x - 2 - 2 = x$
Isolate x Negative Case: Combine like terms by adding $2$ to both sides: $\newline$ $-3x - 4 = x$
Isolate $x$ Negative Case: Combine like terms by adding $2$ to both sides: $\newline$ $-3x - 4 = x$ Add $3x$ to both sides to solve for $x$ : $\newline$ $-4 = 4x$
Isolate x Negative Case: Combine like terms by adding $2$ to both sides: $\newline$ $-3x - 4 = x$ Add $3x$ to both sides to solve for $x$ : $\newline$ $-4 = 4x$ Divide both sides by $4$ to isolate $x$ : $\newline$ $x = -4 / 4$ $\newline$ $x = -1$ $\newline$ This is the solution for the case where $3x + 2$ is negative.