To determine which two given expressions are equivalent to [tex]\(-2 \frac{4}{9}\)[/tex], let's carefully break down and evaluate each expression step-by-step.
First, convert the mixed number [tex]\(-2 \frac{4}{9}\)[/tex] to an improper fraction or a decimal:
- [tex]\(-2 \frac{4}{9}\)[/tex] can be written as [tex]\(-2 - \frac{4}{9} \approx -2.4444\)[/tex].
Now, we will evaluate each expression:
1. Expression 1: [tex]\(-1 \left(\frac{4}{9} + \frac{4}{9}\right)\)[/tex]
- First, compute the sum inside the parenthesis: [tex]\(\frac{4}{9} + \frac{4}{9} = \frac{8}{9}\)[/tex].
- Then, multiply by [tex]\(-1\)[/tex]: [tex]\(-1 \left(\frac{8}{9}\right) = -\frac{8}{9} \approx -0.8889\)[/tex].
- This does not equal [tex]\(-2 \frac{4}{9}\)[/tex].
2. Expression 2: [tex]\(-1 \left(2 + \frac{4}{9}\right)\)[/tex]
- First, compute the sum inside the parenthesis: [tex]\(2 + \frac{4}{9} = 2 \frac{4}{9} \approx 2.4444\)[/tex].
- Then, multiply by [tex]\(-1\)[/tex]: [tex]\(-1 \left(2.4444\right) = -2.4444\)[/tex].
- This is equal to [tex]\(-2 \frac{4}{9}\)[/tex].
3. Expression 3: [tex]\(-2 + \left(-\frac{4}{9}\right)\)[/tex]
- Convert to a single addition operation: [tex]\(-2 - \frac{4}{9} \approx -2 - 0.4444 \approx -2.4444\)[/tex].
- This is equal to [tex]\(-2 \frac{4}{9}\)[/tex].
4. Expression 4: [tex]\(-2 + \frac{4}{9}\)[/tex]
- Perform the addition operation: [tex]\(-2 + \frac{4}{9} \approx -2 + 0.4444 \approx -1.5556\)[/tex].
- This does not equal [tex]\(-2 \frac{4}{9}\)[/tex].
5. Expression 5: [tex]\(2 + \frac{4}{9}\)[/tex]
- Perform the addition operation: [tex]\(2 + \frac{4}{9} \approx 2 + 0.4444 \approx 2.4444\)[/tex].
- This does not equal [tex]\(-2 \frac{4}{9}\)[/tex].
The expressions that are equivalent to [tex]\(-2 \frac{4}{9}\)[/tex] are:
- [tex]\(\boxed{-1 \left(2 + \frac{4}{9}\right)}\)[/tex]
- [tex]\(\boxed{-2 + \left(-\frac{4}{9}\right)}\)[/tex]
