91Ó°ÊÓ

Solving equations, especially those involving absolute values, might seem tricky at first, but it's quite systematic. Let's break it down. When we encounter an equation like \[|3(x-5)+2|+3=9\]our goal is to solve for \(x\). The first step involves isolating the absolute value expression. This means we need to get \[|3(x-5)+2|\]all by itself on one side of the equation. To do this, subtract 3 from both sides, giving us:\[|3(x-5)+2| = 6\]This isolation helps us focus on the absolute value itself. Without it, we're still burdened by the additional terms. Once the absolute value is alone, we can approach the next step, which involves exploring the values that satisfy this expression.

Algebraic manipulation is the secret sauce that helps solve equations, especially when dealing with absolute values. With the isolated absolute value, which is now \[|3(x-5)+2| = 6\]we convert this into two separate equations. You see, an absolute value equation, like \(|A| = B\), is saying either \(A = B\) or \(A = -B\). Thus, we have:- \(3(x-5)+2 = 6\)- \(3(x-5)+2 = -6\)For each equation, we perform operations to isolate \(x\).

• Subtract 2 from both sides.
• Divide by 3 to address the multiplication, aiming to solve for \(x\).

The first equation transforms step-by-step to \(x = \frac{19}{3}\), and the second becomes \(x = \frac{7}{3}\). These operations show how algebraic manipulation simplifies and solves absolute value equations.

Verification of solutions is crucial for all types of equations, especially when absolute values are involved. After obtaining potential solutions, \(x = \frac{19}{3}\) and \(x = \frac{7}{3}\), we must ensure they actually satisfy the initial equation. To verify, substitute each value back into the original equation:- Start with \(x = \frac{19}{3}\), substitute it in, and confirm that both sides balance to 9.- Then, substitute \(x = \frac{7}{3}\) and perform the same check.This process might seem tedious, but it's necessary to confirm the validity of each solution. If both solutions make the left-hand side equal to 9, as intended, then you've successfully verified that they are correct. Always remember, verifying solutions avoids errors in calculations and ensures a thorough understanding of the problem's demands.