Certainly! Let's break down the expression step-by-step and understand how to solve it:
Given the expression:
[tex]\[ \frac{15 \div 8}{\left(\frac{41}{3} \times \frac{8}{4}\right)} \][/tex]
We'll solve it step by step:
1. Calculate the multiplication in the denominator:
[tex]\[ \frac{41}{3} \times \frac{8}{4} \][/tex]
First, simplify [tex]\(\frac{8}{4}\)[/tex]:
[tex]\[ \frac{8}{4} = 2 \][/tex]
Now multiply it by [tex]\(\frac{41}{3}\)[/tex]:
[tex]\[ \frac{41}{3} \times 2 = \frac{41 \times 2}{3} = \frac{82}{3} \][/tex]
2. Now perform the division in the numerator:
[tex]\[ \frac{15}{8} \div \frac{82}{3} \][/tex]
Division by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[ \frac{15}{8} \div \frac{82}{3} = \frac{15}{8} \times \frac{3}{82} \][/tex]
Simplify:
[tex]\[ \frac{15 \times 3}{8 \times 82} = \frac{45}{656} \][/tex]
Now, we need to convert this fraction to its decimal form to match the required format:
[tex]\[ \frac{45}{656} \approx 0.06859756097560976 \][/tex]
This numerical result [tex]\(0.06859756097560976\)[/tex] is obtained.
For the second part of the question:
We're asked for which value of [tex]\(a\)[/tex] does [tex]\(|x| = a\)[/tex] have two solutions.
The equation [tex]\(|x| = a\)[/tex] has two solutions when [tex]\(a > 0\)[/tex]. The absolute value of [tex]\(x\)[/tex] equates to a positive number [tex]\(a\)[/tex] since [tex]\(|x| = a\)[/tex] yields [tex]\(x = a\)[/tex] or [tex]\(x = -a\)[/tex].
Therefore, [tex]\(a\)[/tex] must be the positive value derived from the previous calculation.
So, [tex]\(a = 0.06859756097560976\)[/tex]. This is the value for which [tex]\(|x| = a\)[/tex] has two solutions.
