Answer :
We begin by rewriting the original expression by combining its constant terms:
$$
8.9x + 6.2 + 8.7 = 8.9x + (6.2 + 8.7) = 8.9x + 14.9.
$$
An equivalent expression must have the same coefficient for $x$ (which is $8.9$) and the same constant term ($14.9$). We now examine the options one by one:
1. Option 1: $$9x + 6 + 9$$
The coefficient of $x$ is $9$ and the constant term is $6 + 9 = 15$. Since $9 \neq 8.9$ and $15 \neq 14.9$, this expression is not equivalent.
2. Option 2: $$8.9 + 6.2 + 8.7x$$
Arranging the terms (reordering does not change the value), we have
$$8.7x + 8.9 + 6.2.$$
Here, the coefficient of $x$ is $8.7$ (not $8.9$) and the constant is $8.9 + 6.2 = 15.1$. Thus, this expression is not equivalent.
3. Option 3: $$8.9x + 8.7 + 6.2$$
Rearranging the constant terms gives
$$8.9x + (8.7 + 6.2) = 8.9x + 14.9.$$
This exactly matches the simplified original expression, so it is equivalent.
4. Option 4: $$8.7 + 8.9x + 6.2$$
Again, grouping the like terms gives
$$8.9x + (8.7 + 6.2) = 8.9x + 14.9.$$
This is equivalent to the original expression.
5. Option 5: $$6.2 + 8.7 + 8.9$$
This expression has no $x$ term and the sum of the constants is
$$6.2 + 8.7 + 8.9 = 23.8,$$
which is not equivalent to $8.9x + 14.9$.
6. Option 6: Involves the expression $\frac{3}{2}+\left(\frac{-11}{2}\right.$, which simplifies to $\frac{3}{2} - \frac{11}{2} = -\frac{8}{2} = -4$. This is clearly not equivalent.
7. Option 7: $$\frac{-11}{2}$$
This equals $-5.5$, which is not equivalent.
8. Option 8: $$-\frac{8}{2} -$$
This expression is incomplete and not equivalent to the original expression.
9. Option 9: $$6.2 + 8.7 + 8.9x$$
Rearranging gives
$$8.9x + (6.2 + 8.7) = 8.9x + 14.9,$$
so this expression is equivalent.
10. Option 10: $$(8.7 + 8.9x) + 6.2$$
Grouping the terms, we have
$$8.9x + (8.7 + 6.2) = 8.9x + 14.9.$$
This is also equivalent.
Thus, the expressions that are equivalent to $$8.9x + 6.2 + 8.7$$ are those in Options 3, 4, 9, and 10.
$$
8.9x + 6.2 + 8.7 = 8.9x + (6.2 + 8.7) = 8.9x + 14.9.
$$
An equivalent expression must have the same coefficient for $x$ (which is $8.9$) and the same constant term ($14.9$). We now examine the options one by one:
1. Option 1: $$9x + 6 + 9$$
The coefficient of $x$ is $9$ and the constant term is $6 + 9 = 15$. Since $9 \neq 8.9$ and $15 \neq 14.9$, this expression is not equivalent.
2. Option 2: $$8.9 + 6.2 + 8.7x$$
Arranging the terms (reordering does not change the value), we have
$$8.7x + 8.9 + 6.2.$$
Here, the coefficient of $x$ is $8.7$ (not $8.9$) and the constant is $8.9 + 6.2 = 15.1$. Thus, this expression is not equivalent.
3. Option 3: $$8.9x + 8.7 + 6.2$$
Rearranging the constant terms gives
$$8.9x + (8.7 + 6.2) = 8.9x + 14.9.$$
This exactly matches the simplified original expression, so it is equivalent.
4. Option 4: $$8.7 + 8.9x + 6.2$$
Again, grouping the like terms gives
$$8.9x + (8.7 + 6.2) = 8.9x + 14.9.$$
This is equivalent to the original expression.
5. Option 5: $$6.2 + 8.7 + 8.9$$
This expression has no $x$ term and the sum of the constants is
$$6.2 + 8.7 + 8.9 = 23.8,$$
which is not equivalent to $8.9x + 14.9$.
6. Option 6: Involves the expression $\frac{3}{2}+\left(\frac{-11}{2}\right.$, which simplifies to $\frac{3}{2} - \frac{11}{2} = -\frac{8}{2} = -4$. This is clearly not equivalent.
7. Option 7: $$\frac{-11}{2}$$
This equals $-5.5$, which is not equivalent.
8. Option 8: $$-\frac{8}{2} -$$
This expression is incomplete and not equivalent to the original expression.
9. Option 9: $$6.2 + 8.7 + 8.9x$$
Rearranging gives
$$8.9x + (6.2 + 8.7) = 8.9x + 14.9,$$
so this expression is equivalent.
10. Option 10: $$(8.7 + 8.9x) + 6.2$$
Grouping the terms, we have
$$8.9x + (8.7 + 6.2) = 8.9x + 14.9.$$
This is also equivalent.
Thus, the expressions that are equivalent to $$8.9x + 6.2 + 8.7$$ are those in Options 3, 4, 9, and 10.
