91Ó°ÊÓ

To solve absolute value equations, the first crucial step is to isolate the absolute value expression. This means you need to get the absolute value term by itself on one side of the equation.
In our example, we start with: \ \ \[ \bigg|\frac{3}{4}x - 3\bigg| + 7 = 2 \] \ \ Here, the absolute value expression \( \bigg|\frac{3}{4}x - 3\bigg| \) is not isolated. To isolate it, you subtract 7 from both sides: \ \ \[ \bigg|\frac{3}{4}x - 3\bigg| + 7 - 7 = 2 - 7 \] \ \ \[ \bigg|\frac{3}{4}x - 3\bigg| = -5 \] \ \ Once isolated, you can observe that the absolute value term \( \bigg|\frac{3}{4}x - 3\bigg| \) equals -5, which leads to the next important point. An absolute value cannot be negative, bringing up the impossibility of finding a solution in this case.

Understanding the properties of absolute values is essential for solving these kinds of equations. The absolute value of a number is always non-negative. This means on a number line, it represents the distance from zero and therefore, it can never be negative.
The property of absolute value can be summarized:

• \( |a| \geq 0 \) for any real number \(a\).
• \( |a| = a \) if \(a\) is positive or zero.
• \( |a| = -a \) if \(a\) is negative.

In the given exercise, after isolating the absolute value, we got: \ \[ \bigg|\frac{3}{4}x - 3\bigg| = -5 \] \ Since \( \bigg|\frac{3}{4}x - 3\bigg| \) must be non-negative, it cannot equal -5. This negative right-hand side means there's no solution for the equation.

Solving absolute value equations often involves breaking down the problem into more manageable parts. Typically, once the absolute value is isolated and properties are applied, you would normally set up two possible equations because \( |a| = b \) translates into: \ \[ a = b \text{ or } a = -b \]
In the original example, if the equation had a non-negative number on the right, you’d proceed by removing the absolute value, setting up two cases, and solving each one separately. These two cases would cover all possible solutions:

• \( \frac{3}{4}x - 3 = b \)
• \( \frac{3}{4}x - 3 = -b \)

However, in our exercise, since the right-hand side is negative, we did not need to go further with this step as we already concluded no solutions would exist due to the absolute value properties.
This shows the importance of comprehending the whole process before attempting to solve absolute value equations. Understanding when to recognize an equation has no solution based on inherent mathematical properties saves time and effort.