Question

Rewrite the equation \frac{5}{2x+8}-1=9 as two equations joined by "and"
\frac{5}{b}=10 and x≠?

Answer 1

To rewrite the given equation \(\frac{5}{2x+8}-1=9\) as two equations joined by "and", we first need to isolate \(\frac{5}{2x+8}\).

1. Start by adding 1 to both sides of the equation:
\[
\frac{5}{2x+8} = 10
\]

2. We can set this equal to \(\frac{5}{b} = 10\) by letting \(b = 2x + 8\):
\[
\frac{5}{b} = 10
\]

3. Now we need to solve for \(x\) in terms of \(b\):
\[
10b = 5 \implies b = \frac{1}{2}
\]

From this, we have:
- \(b = 2x + 8\)
- Substituting \(b = \frac{1}{2}\):
\[
2x + 8 = \frac{1}{2}
\]
\[
2x = \frac{1}{2} - 8 = \frac{1}{2} - \frac{16}{2} = -\frac{15}{2}
\]
\[
x = -\frac{15}{4}
\]

4. Thus, \(x\) must be unequal to any value that would make \(2x + 8 = 0\), since that would make the original equation undefined:
\[
2x + 8 = 0 \implies 2x = -8 \implies x = -4
\]

Now, we can express the original equation as two statements joined by "and":
\[
\frac{5}{b} = 10 \text{ and } x \neq -4
\]

Final answer:
\[
\frac{5}{b}=10 \text{ and } x \neq -4
\]