Answer :
To determine which expressions are equivalent to [tex]\(8.9x + 6.2 + 8.7\)[/tex], let's analyze the options by rearranging and simplifying where necessary.
1. [tex]\(9x + 6 + 9\)[/tex]
This expression is not equivalent because the coefficients and constant terms do not match the original expression.
2. [tex]\(8.9 + 6.2 + 8.7x\)[/tex]
This expression changes the order of addition, but the terms themselves (coefficients and constants) are not correctly rearranged. It places 8.9 as a constant and [tex]\(8.7x\)[/tex] as the variable term, which is not equivalent.
3. [tex]\(8.9x + 8.7 + 6.2\)[/tex]
This expression is equivalent to the original because it simply changes the order of the constant terms. The commutative property of addition allows us to rearrange the constants: [tex]\(8.9x + 6.2 + 8.7\)[/tex] can be reordered as [tex]\(8.9x + 8.7 + 6.2\)[/tex].
4. [tex]\(8.7 + 8.9x + 6.2\)[/tex]
This expression is equivalent as well, rearranging the same terms: [tex]\(8.9x + 6.2 + 8.7\)[/tex] can be rearranged to [tex]\(8.7 + 8.9x + 6.2\)[/tex], which still maintains the same sum due to the commutative property.
5. [tex]\(6.2 + 8.7 + 8.9\)[/tex]
This expression has no variable term [tex]\(x\)[/tex], making it unequal because it sums constants only, missing the term with [tex]\(x\)[/tex].
6. [tex]\(6.2 + 8.7 + 8.9x\)[/tex]
This is equivalent to the original expression. Reordering, the constants [tex]\(6.2\)[/tex] and [tex]\(8.7\)[/tex] get summed with [tex]\(8.9x\)[/tex], i.e., [tex]\(8.9x + 6.2 + 8.7\)[/tex] can become [tex]\(6.2 + 8.7 + 8.9x\)[/tex].
7. [tex]\(0.9 + 8.2x + 8.7\)[/tex]
This is not equivalent because the coefficients and terms do not match. Here, both the [tex]\(x\)[/tex] term and the constant term are incorrect.
The equivalent expressions from the provided options are:
- [tex]\(8.9x + 8.7 + 6.2\)[/tex]
- [tex]\(8.7 + 8.9x + 6.2\)[/tex]
- [tex]\(6.2 + 8.7 + 8.9x\)[/tex]
Thus, the expressions that are equivalent to [tex]\(8.9x + 6.2 + 8.7\)[/tex] are options 3, 4, and 6.
1. [tex]\(9x + 6 + 9\)[/tex]
This expression is not equivalent because the coefficients and constant terms do not match the original expression.
2. [tex]\(8.9 + 6.2 + 8.7x\)[/tex]
This expression changes the order of addition, but the terms themselves (coefficients and constants) are not correctly rearranged. It places 8.9 as a constant and [tex]\(8.7x\)[/tex] as the variable term, which is not equivalent.
3. [tex]\(8.9x + 8.7 + 6.2\)[/tex]
This expression is equivalent to the original because it simply changes the order of the constant terms. The commutative property of addition allows us to rearrange the constants: [tex]\(8.9x + 6.2 + 8.7\)[/tex] can be reordered as [tex]\(8.9x + 8.7 + 6.2\)[/tex].
4. [tex]\(8.7 + 8.9x + 6.2\)[/tex]
This expression is equivalent as well, rearranging the same terms: [tex]\(8.9x + 6.2 + 8.7\)[/tex] can be rearranged to [tex]\(8.7 + 8.9x + 6.2\)[/tex], which still maintains the same sum due to the commutative property.
5. [tex]\(6.2 + 8.7 + 8.9\)[/tex]
This expression has no variable term [tex]\(x\)[/tex], making it unequal because it sums constants only, missing the term with [tex]\(x\)[/tex].
6. [tex]\(6.2 + 8.7 + 8.9x\)[/tex]
This is equivalent to the original expression. Reordering, the constants [tex]\(6.2\)[/tex] and [tex]\(8.7\)[/tex] get summed with [tex]\(8.9x\)[/tex], i.e., [tex]\(8.9x + 6.2 + 8.7\)[/tex] can become [tex]\(6.2 + 8.7 + 8.9x\)[/tex].
7. [tex]\(0.9 + 8.2x + 8.7\)[/tex]
This is not equivalent because the coefficients and terms do not match. Here, both the [tex]\(x\)[/tex] term and the constant term are incorrect.
The equivalent expressions from the provided options are:
- [tex]\(8.9x + 8.7 + 6.2\)[/tex]
- [tex]\(8.7 + 8.9x + 6.2\)[/tex]
- [tex]\(6.2 + 8.7 + 8.9x\)[/tex]
Thus, the expressions that are equivalent to [tex]\(8.9x + 6.2 + 8.7\)[/tex] are options 3, 4, and 6.
