Let's match each phrase to the correct algebraic expression step-by-step:
### Phrase 1: Five less than the quotient of eight and a number increased by seven
1. Define the number as [tex]\( x \)[/tex].
2. The statement "the quotient of eight and a number" translates to [tex]\( \frac{8}{x} \)[/tex].
3. "Increased by seven" means we add 7 to [tex]\( \frac{8}{x} \)[/tex], which gives [tex]\( \frac{8}{x} + 7 \)[/tex].
4. Finally, "five less than" indicates we subtract 5 from [tex]\( \frac{8}{x} + 7 \)[/tex], which results in [tex]\( \frac{8}{x} + 7 - 5 \)[/tex].
5. Simplifying this expression, we get [tex]\( \frac{8}{x} + 2 \)[/tex].
So, the expression is [tex]\( \frac{8}{x} + 2 \)[/tex].
### Phrase 2: Twice the difference of eight and a number
1. Define the number as [tex]\( x \)[/tex].
2. The statement "the difference of eight and a number" is described by [tex]\( 8 - x \)[/tex].
3. "Twice" means we multiply the difference by 2, resulting in [tex]\( 2 \times (8 - x) \)[/tex].
So, the expression is [tex]\( 2 \times (8 - x) \)[/tex].
### Phrase 3: Eight more than the product of seven and a number decreased by two
1. Define the number as [tex]\( x \)[/tex].
2. "The product of seven and a number" translates to [tex]\( 7x \)[/tex].
3. "Decreased by two" indicates we need to subtract 2 from [tex]\( x \)[/tex], giving us [tex]\( 7 \times (x - 2) \)[/tex].
4. Finally, "eight more than" means we add 8 to the product, resulting in [tex]\( 8 + 7 \times (x - 2) \)[/tex].
So, the expression is [tex]\( 8 + 7 \times (x - 2) \)[/tex].
### Matching Summary:
1. Five less than the quotient of eight and a number increased by seven is matched with [tex]\( \frac{8}{x} + 2 \)[/tex].
2. Twice the difference of eight and a number is matched with [tex]\( 2 \times (8 - x) \)[/tex].
3. Eight more than the product of seven and a number decreased by two is matched with [tex]\( 8 + 7 \times (x - 2) \)[/tex].
Hence, these phrases correspond to the algebraic expressions as described.
