Questions: Determine whether the given value is a solution of the equation. 2-[3-(3-x)]=3x-(4+x) (a) x=2 Yes No (b) x=4 Yes No

Transcript text: Determine whether the given value is a solution of the equation. \[ 2-[3-(3-x)]=3 x-(4+x) \] (a) $x=2$ Yes No (b) $x=4$ Yes No

Solution

Solution Steps

Step 1: Simplify the equation

First, simplify the given equation: \[ 2 - [3 - (3 - x)] = 3x - (4 + x) \]

Simplify the left-hand side (LHS): \[ 2 - [3 - 3 + x] = 2 - [x] = 2 - x \]

Simplify the right-hand side (RHS): \[ 3x - 4 - x = 2x - 4 \]

Now the equation becomes: \[ 2 - x = 2x - 4 \]

Step 2: Solve for $x$

Combine like terms: \[ 2 + 4 = 2x + x \] \[ 6 = 3x \] \[ x = 2 \]

Step 3: Verify the solution for $x = 2$

Substitute $x = 2$ into the original equation: \[ 2 - [3 - (3 - 2)] = 3(2) - (4 + 2) \] \[ 2 - [3 - 1] = 6 - 6 \] \[ 2 - 2 = 0 \] \[ 0 = 0 \] The equation holds true, so $x = 2$ is a solution.

Step 4: Verify the solution for $x = 4$

Substitute $x = 4$ into the original equation: \[ 2 - [3 - (3 - 4)] = 3(4) - (4 + 4) \] \[ 2 - [3 - (-1)] = 12 - 8 \] \[ 2 - 4 = 4 \] \[ -2 = 4 \] The equation does not hold true, so $x = 4$ is not a solution.

Final Answer

(a) $x = 2$: $\boxed{\text{Yes}}$

(b) $x = 4$: $\boxed{\text{No}}$

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