Answer :
Let's solve the problem step by step.
First Expression Problem:
We need to find which among the given options are equivalent to the expression [tex]\(8 \frac{1}{2} x + \frac{1}{2} + 3 \frac{1}{2} x - 2 x\)[/tex].
We start by simplifying the given expression:
1. Convert mixed numbers to improper fractions if necessary.
2. Combine the like terms.
The expression is:
[tex]\[ 8\frac{1}{2}x + \frac{1}{2} + 3\frac{1}{2}x - 2x \][/tex]
First, combine the [tex]\( x \)[/tex]-terms:
- [tex]\( 8\frac{1}{2}x \)[/tex] is [tex]\( 8.5x = \frac{17}{2}x \)[/tex]
- [tex]\( 3\frac{1}{2}x \)[/tex] is [tex]\( 3.5x = \frac{7}{2}x \)[/tex]
- Combine: [tex]\( \frac{17}{2}x + \frac{7}{2}x - 2x \)[/tex]
The [tex]\( x \)[/tex]-terms become:
[tex]\[ \frac{17}{2}x + \frac{7}{2}x = \frac{24}{2}x = 12x \][/tex]
Subtract [tex]\( 2x \)[/tex]:
[tex]\[ 12x - 2x = 10x \][/tex]
Now for the constant terms:
- We have [tex]\( \frac{1}{2} \)[/tex].
Combine all:
[tex]\[ 10x + \frac{1}{2} \][/tex]
Now, let's check which options match:
- [tex]\( \boxed{10x + \frac{1}{2}} \)[/tex]
Second Expression Problem:
For the expression [tex]\(\frac{1}{2} x + 4 \frac{1}{2} + \frac{1}{2} x - \frac{1}{2}\)[/tex], let's simplify it:
1. Combine the [tex]\( x \)[/tex]-terms:
[tex]\( \frac{1}{2}x + \frac{1}{2}x = x \)[/tex]
2. Simplify the constant terms:
- [tex]\( 4\frac{1}{2} \)[/tex] is [tex]\( 4.5 \)[/tex]
- [tex]\( 4.5 - \frac{1}{2} = 4 \)[/tex]
Putting it back together, we get:
[tex]\[ x + 4 \][/tex]
Among the options provided, [tex]\( \boxed{x + 4} \)[/tex] is equivalent.
So, the solutions are:
- For the first expression, the equivalent expression is [tex]\( 10x + \frac{1}{2} \)[/tex].
- For the second expression, the equivalent expression is [tex]\( x + 4 \)[/tex].
First Expression Problem:
We need to find which among the given options are equivalent to the expression [tex]\(8 \frac{1}{2} x + \frac{1}{2} + 3 \frac{1}{2} x - 2 x\)[/tex].
We start by simplifying the given expression:
1. Convert mixed numbers to improper fractions if necessary.
2. Combine the like terms.
The expression is:
[tex]\[ 8\frac{1}{2}x + \frac{1}{2} + 3\frac{1}{2}x - 2x \][/tex]
First, combine the [tex]\( x \)[/tex]-terms:
- [tex]\( 8\frac{1}{2}x \)[/tex] is [tex]\( 8.5x = \frac{17}{2}x \)[/tex]
- [tex]\( 3\frac{1}{2}x \)[/tex] is [tex]\( 3.5x = \frac{7}{2}x \)[/tex]
- Combine: [tex]\( \frac{17}{2}x + \frac{7}{2}x - 2x \)[/tex]
The [tex]\( x \)[/tex]-terms become:
[tex]\[ \frac{17}{2}x + \frac{7}{2}x = \frac{24}{2}x = 12x \][/tex]
Subtract [tex]\( 2x \)[/tex]:
[tex]\[ 12x - 2x = 10x \][/tex]
Now for the constant terms:
- We have [tex]\( \frac{1}{2} \)[/tex].
Combine all:
[tex]\[ 10x + \frac{1}{2} \][/tex]
Now, let's check which options match:
- [tex]\( \boxed{10x + \frac{1}{2}} \)[/tex]
Second Expression Problem:
For the expression [tex]\(\frac{1}{2} x + 4 \frac{1}{2} + \frac{1}{2} x - \frac{1}{2}\)[/tex], let's simplify it:
1. Combine the [tex]\( x \)[/tex]-terms:
[tex]\( \frac{1}{2}x + \frac{1}{2}x = x \)[/tex]
2. Simplify the constant terms:
- [tex]\( 4\frac{1}{2} \)[/tex] is [tex]\( 4.5 \)[/tex]
- [tex]\( 4.5 - \frac{1}{2} = 4 \)[/tex]
Putting it back together, we get:
[tex]\[ x + 4 \][/tex]
Among the options provided, [tex]\( \boxed{x + 4} \)[/tex] is equivalent.
So, the solutions are:
- For the first expression, the equivalent expression is [tex]\( 10x + \frac{1}{2} \)[/tex].
- For the second expression, the equivalent expression is [tex]\( x + 4 \)[/tex].
