91Ó°ÊÓ

Solving equations is all about finding the value of unknown variables that make an equation true. In the context of absolute value equations, such as \( \left|\frac{6-x}{4}\right|=5 \), this process can be a bit more complex due to the nature of absolute values. To find the solutions, we need to consider all possible scenarios that meet the equation's condition. The equation essentially sets the expression inside the absolute value bars equal to both the positive and negative of the other side. This approach comes from the fact that absolute values measure the distance from zero on a number line, making negative distances also positive. For our equation, this results in:

• \( \frac{6-x}{4}=5 \)
• \( \frac{6-x}{4}=-5 \)

This ensures we capture all potential solutions. Each of these resulting equations is then solved independently to find the possible values of \(x\). Solving both equations and combining their results gives the complete solution to the original absolute value equation.

Algebraic manipulation involves using algebraic methods to simplify or rearrange equations to make them easier to solve. This often includes tasks such as getting rid of fractions, isolating variables, and performing arithmetic operations. In solving the absolute value equation \( \left|\frac{6-x}{4}\right|=5 \), manipulation is a key part of finding the solutions.For each condition derived from the absolute value equation, the first step is to eliminate fractions. We do this by multiplying each side by the denominator, in this case, the number 4. Consider:

• Starting with \( \frac{6-x}{4} = 5 \), multiply by 4 to get \( 6-x = 20 \).
• Similarly, for \( \frac{6-x}{4} = -5 \), multiply by 4 to obtain \( 6-x = -20 \).

The next step is to isolate the variable \(x\). By rearranging terms and solving the resulting linear equations, we find \(x\) values that solve the problem. Algebraic manipulation continues as we ensure the variable is alone on one side, leading us to the potential solutions: \(x = -14\) and \(x = 26\).

Absolute value properties are fundamental in understanding how to work with equations involving absolute values. The absolute value of a number is its distance from zero, ignoring any plus or minus sign. This property is why absolute value equations split into two separate equations — because a number and its negative both have the same distance from zero. In our example, with the equation \( \left|\frac{6-x}{4}\right|=5 \), we set the expression \( \frac{6-x}{4} \) equal to both 5 and -5. These new equations represent both possible distances from zero, effectively representing all scenarios where the original absolute value equation holds true. The steps outlined highlight the necessity of considering both positive and negative solutions when dealing with absolute values, showcasing their unique property that alters the equation-solving process.This is a key reason why absolute value problems often have two potential solutions, covering all possible cases where the original expression's value could yield the target magnitude (such as the 5 in this exercise). Understanding these properties makes it easier to predict how an absolute value equation might behave and ensures you solve it comprehensively.