91Ó°ÊÓ

When solving absolute value equations, the first step is to isolate the absolute value expression. This means you want the absolute value term all by itself on one side of the equation.
In the example given, we start with the equation: \[5|7 - 3x| + 2 = 4|7 - 3x| - 1\]First, we move the absolute value terms to one side of the equation by subtracting \(4|7 - 3x|\) from both sides:
\[5|7 - 3x| - 4|7 - 3x| + 2 = -1\]This simplifies to:
\[|7 - 3x| + 2 = -1\]The absolute value term, \(|7 - 3x|\), is now isolated on the left side. Isolating the term helps in understanding and solving the equation correctly.
A key thing to remember is to carefully perform operations that allow you to get the absolute value term by itself.

An important property of absolute value is that it is always non-negative. This means that an absolute value expression like \(|7 - 3x|\) can never be less than zero.
In our equation: \[|7 - 3x| + 2 = -1\]After isolating the absolute value term and simplifying the equation, we notice that \(|7 - 3x| + 2\) equals \(-1\). Because \(|7 - 3x|\) is always non-negative (\(0\) or greater), adding 2 to it will always produce a result that is at least 2.
So, it is impossible for \(|7 - 3x| + 2\) to equal \(-1\). This highlights the non-negative property of absolute values and is crucial to understand while solving such equations.
Recognizing the non-negative property helps draw conclusions about the feasibility of the equation.

When an equation involving an absolute value results in a contradiction, such as a negative number on one side of the equation, it implies that there is no possible solution.
In our example, we reached the equation:\[|7 - 3x| = -3\]We know absolute values cannot be negative (\(|7 - 3x| \geq 0)\), so it’s not possible for \(|7 - 3x|\) to equal \(-3\).
This contradiction means there is no value of \(x\) that will satisfy the equation. Therefore, the original equation: \[5|7 - 3x| + 2 = 4|7 - 3x| - 1\]has no solution.
Understanding situations where an equation has no solution is crucial as it helps save time and avoid unnecessary calculations.