Let’s solve the equation [tex]\(-2x + 6 = -\left(\frac{2}{3}\right)^2 + 5\)[/tex] using the given options and approximate the solution.
1. Simplify the right-hand side:
- Calculate [tex]\(\left(\frac{2}{3}\right)^2\)[/tex]:
[tex]\[
\left(\frac{2}{3}\right)^2 = \frac{4}{9}
\][/tex]
- Add this value to the equation:
[tex]\[
-\left(\frac{4}{9}\right) + 5 = -\frac{4}{9} + 5 = 5 - \frac{4}{9}
\][/tex]
- Express [tex]\(5\)[/tex] as a fraction with denominator [tex]\(9\)[/tex]:
[tex]\[
5 = \frac{45}{9}
\][/tex]
- Simplify:
[tex]\[
5 - \frac{4}{9} = \frac{45}{9} - \frac{4}{9} = \frac{41}{9}
\][/tex]
So, the equation simplifies to:
[tex]\[
-2x + 6 = \frac{41}{9}
\][/tex]
2. Isolate [tex]\(x\)[/tex]:
- Subtract 6 from both sides:
[tex]\[
-2x = \frac{41}{9} - 6
\][/tex]
- Convert [tex]\(6\)[/tex] to a fraction with denominator [tex]\(9\)[/tex]:
[tex]\[
6 = \frac{54}{9}
\][/tex]
- Subtract:
[tex]\[
\frac{41}{9} - \frac{54}{9} = \frac{41 - 54}{9} = \frac{-13}{9}
\][/tex]
So, we have:
[tex]\[
-2x = \frac{-13}{9}
\][/tex]
- Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
x = \frac{-13}{9} \div -2 = \frac{-13}{9} \times \frac{-1}{2} = \frac{13}{18}
\][/tex]
3. Approximate [tex]\(\frac{13}{18}\)[/tex] using three iterations of successive approximation:
- Convert [tex]\(\frac{13}{18}\)[/tex] to a decimal to better compare it with given options:
[tex]\[
\frac{13}{18} \approx 0.722
\][/tex]
Let’s compare the approximate value [tex]\(0.722\)[/tex] with the given options:
- Option A: [tex]\(\frac{7}{8} = 0.875\)[/tex]
- Option B: [tex]\(\frac{13}{16} = 0.8125\)[/tex]
- Option C: [tex]\(\frac{3}{4} = 0.75\)[/tex]
- Option D: [tex]\(\frac{15}{16} = 0.9375\)[/tex]
Given [tex]\(\frac{13}{18} \approx 0.722\)[/tex], none of the provided options match closely. Since none of the options approximate [tex]\(\frac{13}{18}\)[/tex] within a reasonable margin, the correct answer to this problem is the following:
```
None
```
