Question

1. solve and state the solution using set notation if possible.

$-6 |x + 3| = -12$
$\frac{-6}{-6}$ $\frac{-12}{-6}$
$x + 3 = 2$
$\frac{-3}{}$ $\frac{-3}{}$
$x = -1$

Explanation:

Step1: Isolate the absolute value

Given the equation \(-6|x + 3|=-12\), divide both sides by \(-6\) to isolate the absolute value expression.
\(\frac{-6|x + 3|}{-6}=\frac{-12}{-6}\)
Simplifying both sides, we get \(|x + 3| = 2\).

Step2: Solve the absolute value equation

The absolute value equation \(|A| = B\) (where \(B\geq0\)) is equivalent to \(A = B\) or \(A=-B\). Here, \(A=x + 3\) and \(B = 2\), so we have two cases:

• Case 1: \(x+3=2\)

Subtract 3 from both sides: \(x=2 - 3=-1\)

• Case 2: \(x + 3=-2\)

Subtract 3 from both sides: \(x=-2-3=-5\)

Step3: Write the solution in set notation

The solutions are \(x=-1\) and \(x = - 5\), so in set notation, the solution set is \(\{-5,-1\}\).

Answer:

\(\{-5, - 1\}\)