Let's solve each part of the question step-by-step.
### Part a:
The equation we are given is:
[tex]\[ x - \frac{1}{2} = \pi \][/tex]
This means that we are trying to solve for [tex]\(x\)[/tex].
1. First, isolate [tex]\(x\)[/tex] by adding [tex]\(\frac{1}{2}\)[/tex] to both sides of the equation:
[tex]\[ x = \pi + \frac{1}{2} \][/tex]
2. Now, let's analyze the variable [tex]\(x\)[/tex]. Since [tex]\(\pi\)[/tex] is an irrational number, adding [tex]\(\frac{1}{2}\)[/tex] (a rational number) to it will still yield an irrational number. Therefore:
The variable [tex]\(x\)[/tex] represents:
- an irrational number.
### Part b:
We know that:
[tex]\[ x = \pi + \frac{1}{2} \][/tex]
To complete the statement:
"This is because the [tex]\(\pi + \frac{1}{2}\)[/tex] can be found equation [tex]\(x - \frac{1}{2} = \pi\)[/tex] can be rewritten as [tex]\(x = \pi + \frac{1}{2}\)[/tex]."
We have to identify the value of [tex]\(\pi + \frac{1}{2}\)[/tex].
We are given:
[tex]\[ x = 3.641592653589793 \][/tex]
This numerical value is the answer we get when [tex]\(\pi\)[/tex] (approximately 3.141592653589793) is added to [tex]\(\frac{1}{2}\)[/tex] (which is approximately 0.5):
[tex]\[ \pi + \frac{1}{2} \approx 3.641592653589793 \][/tex]
Now, consider this:
In the statement:
"The value of [tex]\(a \cdot c\)[/tex] is [tex]\(3.641592653589793\)[/tex]",
here [tex]\( a = \pi \)[/tex] (approximately 3.141592653589793) and [tex]\(c = \frac{1}{2} \)[/tex]. When you add these:
[tex]\[ \pi \approx 3.141592653589793 \][/tex]
[tex]\[ \frac{1}{2} \approx 0.5 \][/tex]
Adding these values gives us [tex]\( a \cdot c \approx 3.641592653589793 \)[/tex].
And:
"In the second part, denote [tex]\(d\)[/tex] as & the value of the denominator, [tex]\( b \cdot d \)[/tex], is 1.":
This means [tex]\(b \cdot d\)[/tex] remains as is, since [tex]\( b \cdot d\)[/tex] doesn't change irrational and rational number combination solving calculations.
Therefore, the answer for Part a is:
- The variable [tex]\(x\)[/tex] represents an irrational number.
And for Part b, it's:
- The value of [tex]\( a \cdot c\)[/tex] is [tex]\(3.641592653589793\)[/tex], whereas, the value of denominator [tex]\( b \cdot d \)[/tex] is 1.
