Question

1. $-4w - 2 = |6w + 20|$

Explanation:

Step1: Analyze the absolute value equation

The equation is \(-4w - 2 = |6w + 20|\). By the definition of absolute value, if \(|A| = B\) (where \(B\geq0\)), then \(A = B\) or \(A=-B\). But here, the left - hand side is \(-4w - 2\), and the right - hand side is non - negative (\(|6w + 20|\geq0\)). So we must have \(-4w - 2\geq0\), which implies \(-4w\geq2\), or \(w\leq-\frac{1}{2}\). And we can also set up the equation \(6w + 20=-4w - 2\) (because if \(|A| = B\) and \(B\geq0\), then \(A = B\) or \(A=-B\); here we consider the case \(A=-B\) since \(B=-4w - 2\) and we know \(B\geq0\))

Step2: Solve the linear equation from Step 1

Start with \(6w+20=-4w - 2\)
Add \(4w\) to both sides of the equation:
\(6w + 4w+20=-4w+4w - 2\)
\(10w + 20=-2\)

Step3: Isolate the term with w

Subtract 20 from both sides:
\(10w+20 - 20=-2 - 20\)
\(10w=-22\)

Step4: Solve for w

Divide both sides by 10:
\(w=\frac{-22}{10}=-\frac{11}{5}=-2.2\)

We should also check if this solution satisfies the original equation.
Left - hand side: \(-4\times(-2.2)-2 = 8.8 - 2=6.8\)
Right - hand side: \(|6\times(-2.2)+20|=|-13.2 + 20|=|6.8| = 6.8\)
So \(w = - 2.2\) (or \(w=-\frac{11}{5}\)) is a valid solution.

Answer:

\(w =-\frac{11}{5}\) (or \(w=-2.2\))