Question

what are the solutions of the following equation?
$-3|x + 9| + 2 = 10|x + 9| + 5$
choose 1 answer:
a $x = -9$ or $x = 9$
b $x = -3$ or $x = 3$
c $x = \dfrac{120}{13}$ or $x = -\dfrac{114}{13}$
d only $x = -\dfrac{3}{13}$
e there are no solutions

Explanation:

Step1: Isolate the absolute value term

Start with the equation \(-3|x + 9| + 2 = 10|x + 9| + 5\). First, we want to get all the absolute value terms on one side. Let's add \(3|x + 9|\) to both sides and subtract 5 from both sides.

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Simplify the left side: \(2 - 5=-3\), and the right side: \(10|x + 9|+3|x + 9| = 13|x + 9|\). So we have \(-3 = 13|x + 9|\).

Step2: Analyze the absolute value equation

Recall that the absolute value of a number, \(|a|\), is always non - negative (i.e., \(|a|\geq0\) for all real numbers \(a\)). So \(13|x + 9|\) is a non - negative number because \(13>0\) and \(|x + 9|\geq0\). The product of a positive number (13) and a non - negative number (\(|x + 9|\)) is non - negative, so \(13|x + 9|\geq0\). But our equation is \(-3=13|x + 9|\), and \(-3<0\). There is no real number \(x\) that can make a non - negative number equal to a negative number.

Answer:

E. There are no solutions