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When faced with an equation involving an absolute value, such as \( |2x| = -x + 4 \), the key is to understand that the absolute value expression represents the distance of a number from zero on the number line, regardless of direction. To solve these equations, split the problem into two separate cases: one where the contents of the absolute value are positive, and one where they are negative.

Here's how to approach it:

• Write down two equations, one assuming the expression inside the absolute value is positive (\(2x = -x + 4\)) and the other assuming it is negative (\(2x = x - 4\)).
• Solve each equation separately as you would with a standard linear equation. The goal is to isolate the variable \(x\).

Keep in mind, when removing the absolute value, it's crucial to consider both possible scenarios, as neglecting one can lead to missing real solutions.

To systematically solve equations, one should follow a set of steps that simplifies the process and ensures accuracy:

Identify the equation type

Knowing whether you're dealing with a linear, quadratic, or another type of equation guides the solving strategy. With absolute value equations, remember the special characteristic that requires forming two separate linear equations.

Isolate variables

Manipulate the equation using arithmetic operations to get the variable \(x\) alone on one side of the equation. For example, to solve \(2x = -x + 4\), add \(x\) to both sides to obtain \(3x = 4\) and then divide by 3 to get \(x = \frac{4}{3}\).

Perform operations

Systematically carry out the necessary operations like addition, subtraction, multiplication, or division, ensuring that you do them on both sides of the equation to maintain balance.

Verifying solutions is the final, crucial step in solving equations. This ensures that the solutions obtained are correct and valid. Use the following process:

• Substitute the found solutions back into the original equation.
• Simplify both sides of the equation to check if they are equal.
• Confirm that each solution does not violate any conditions from the original problem, such as non-negativity in the context of absolute values.

For instance, when checking \(x = \frac{4}{3}\) against \( |2x| = -x + 4 \), it holds true as \( |2 * \frac{4}{3}| = -\frac{4}{3} + 4 \), simplifying to \( \frac{8}{3} = \frac{8}{3} \). However, for \(x = -4\), the check \( |2 * -4| = -(-4) + 4 \) fails since \( 8 eq 0 \). Therefore, \(x = -4\) is not a valid solution.