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When solving an absolute value equation, the first step is to isolate the absolute value expression on one side of the equation. This process helps us set the stage for breaking down the equation into simpler parts. For the equation \(5 = |2x| - 3\), isolating the absolute value is crucial.

Consider the isolated equation as a balance. You want to make sure the absolute value expression stands alone without any numbers or operations around it. In this case, we accomplish this by adding 3 to both sides (like moving weights to balance the scales). This gives us

• \(5 + 3 = |2x| - 3 + 3\)
• \(8 = |2x|\)

This clean separation allows you to focus on what's inside the absolute value symbol without distractions. Remember, isolating is the launching point for solving any absolute value equation.

After isolating the absolute value, the next step is to solve the resulting equations. With \(8 = |2x|\), we can break it down further. Absolute value indicates that the expression inside could be either positive or negative and yet produce the same absolute value.

To solve, consider two scenarios:

• \(|2x| = 8\) translates to \(2x = 8\)
• \(|2x| = 8\) also translates to \(2x = -8\)

Now, we have two separate linear equations to solve. Solving these individually will yield simple expressions for \(x\). First, for \(2x = 8\):

• Divide both sides by 2, obtaining \(x = 4\)

Next, for \(2x = -8\):

• Similarly, divide by 2 to get \(x = -4\)

Solving the linear equations provides the values of \(x\) which are possible solutions to the original absolute value equation.

The concept of analyzing both positive and negative cases is central to solving absolute value equations. The absolute value expression \(|2x|\) implies the number \(2x\) is \(8\) units away from \(0\) on the number line.

In mathematical terms, this gives you two possibilities:

• The distance of \(2x\) from \(0\) could be exactly \(8\), giving \(2x = 8\)
• Or \(2x\) is at \(-8\), in the opposite direction, hence \(2x = -8\)

This duality is why an absolute value simplifies into two separate potential equations or cases. By nurturing this habit of considering both cases, you ensure no potential solution is overlooked, affirming both results of \(x = 4\) and \(x = -4\) are correct.
Always remember: Absolute value equals distance in math. It doesn't concern itself with direction, only how far. That’s why checking both possibilities (positive and negative) is indispensable when you deal with absolute values.