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Absolute value equations can be tricky at first, but they're easier to handle once you understand the basic steps. These types of equations involve expressions wrapped within absolute value symbols. The absolute value of a number is always a non-negative value, representing the distance from zero on a number line. Solving such equations generally involves isolating the absolute value expression and then dealing with the two possible scenarios.

Let's begin by noting that the goal of solving any equation is to find the value(s) of the variable that make the equation true. For absolute value equations, isolation of the absolute value term is often the first step. Once isolated, the equation is split into potential scenarios, as each result inside the absolute value can be either positive or negative, leading to two different equations to solve.

When both equations are solved, you obtain the possible solutions for the equation. In many cases, these solutions need to be checked back in the original equation to ensure they satisfy the absolute value condition.

Isolating terms is an essential skill in algebra, especially when dealing with absolute value equations. When you want to solve an equation involving absolute values, the first step is to isolate the absolute value term on one side of the equation. Let's walk through this process.

Consider the equation given in the exercise: \(6|1-2x|-7=11\). The target is to get \(|1-2x|\) by itself. This involves a few operations:

• Add or subtract: Remove any constants added or subtracted from the absolute term. For our example, add 7 to both sides to get \(6|1-2x|=18\).
• Division or multiplication: If there's a coefficient in front of the absolute term, as with the 6 in this example, divide both sides by this number. Doing this simplifies the equation to \(|1-2x|=3\).

With the absolute value term isolated, you are ready for the next step of splitting into two equations based on definition.

After isolating the absolute value, the next step to solving the equation is to split it into two separate equations. The absolute value equation \(|A| = B\) implies two potential scenarios: \(A = B\) and \(A = -B\). This reflects that the expression inside the absolute value can result in either the positive or the negative value.

Taking our example, with \(|1-2x|=3\), we can split it into the following two equations:

• \(1-2x = 3\), which reflects the expression when it equals the positive value.
• \(1-2x = -3\), representing the case when the expression is equivalent to the negative value of absolute value.

Next, solve each equation individually:

For the equation \(1-2x = 3\): subtract 1, and then divide by -2 to find \(x = -1\).
For the equation \(1-2x = -3\): similarly, subtract 1, and divide by -2 to find \(x = 2\).
Therefore, the solutions to the original equation are \(x = -1\) and \(x = 2\). Always make sure to check both solutions to ensure they satisfy the original condition.