What is the value of x in both a & b?

Answer:
I've pretty much already solved this, in my discussion above:
| 3 | = 3
| –3 | = 3
So then x must be equal to 3 or equal to –3.
But how am I supposed to solve this if I don't already know the answer? I will use the positive / negative property of the absolute value to split the equation into two cases, and I will use the fact that the "minus" sign in the negative case indicates "the opposite sign", not "a negative number".
For example, if I have x = –6, then "–x " indicates "the opposite of x" or, in this case, –(–6) = +6, a positive number. The "minus" sign in "–x" just indicates that I am changing the sign on x. It does not indicate a negative number. This distinction is crucial!
Whatever the value of x might be, taking the absolute value of x makes it positive. Since x might originally have been positive and might originally have been negative, I must acknowledge this fact when I remove the absolute-value bars. I do this by splitting the equation into two cases. For this exercise, these cases are as follows:
a. If the value of x was non-negative (that is, if it was positive or zero) to start with, then I can bring that value out of the absolute-value bars without changing its sign, giving me the equation x = 3.
b. If the value of x was negative to start with, then I can bring that value out of the absolute-value bars by changing the sign on x, giving me the equation –x = 3, which solves as x = –3.
Then my solution
Answer:
7 and [tex]\frac{16}{5}[/tex]
Step-by-step explanation:
(a)
Δ ABC and Δ ADE are similar thus the ratios of corresponding sides are equal
[tex]\frac{AB}{AD}[/tex] = [tex]\frac{AC}{AE}[/tex] , substitute values
[tex]\frac{2}{4}[/tex] = [tex]\frac{x}{x+7}[/tex] ( cross- multiply )
4x = 2(x + 7) ← distribute
4x = 2x + 14 ( subtract 2x from both sides )
2x = 14 ( divide both sides by 2 )
x = 7
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(b)
Δ ABC and Δ CDF are similar thus ratios of corresponding sides are equal
[tex]\frac{AB}{CD}[/tex] = [tex]\frac{BC}{DF}[/tex] , substitute values
[tex]\frac{2}{5}[/tex] = [tex]\frac{x}{8}[/tex] ( cross- multiply )
5x = 16 ( divide both sides by 5 )
x = [tex]\frac{16}{5}[/tex]