Variable Expressions in Denominators Quick Check

Question

1 of 51 of 5 Items
Question
Rewrite the equation 52x−8−1=9 as two equations joined by “and.”(1 point)
Responses

52x−8=9 and x≠−8
Start Fraction 5 over 2 x minus 8 End Fraction equals 9 and x does not equal negative 8

52x−8−1=9 and x≠−4
Start Fraction 5 over 2 x minus 8 End Fraction minus 1 equals 9 and x does not equal negative 4

52x−8−1=9 and x≠4
Start Fraction 5 over 2 x minus 8 End Fraction minus 1 equals 9 and x does not equal 4

52x−8−1=9 and x≠8
Start Fraction 5 over 2 x minus 8 End Fraction minus 1 equals 9 and x does not equal 8
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Answer 1

To rewrite the equation \( \frac{5}{2}x - 8 - 1 = 9 \) as two equations joined by “and,” we first need to isolate the fraction and determine its domain.

1. Start with the original equation: \[ \frac{5}{2}x - 8 - 1 = 9 \]

2. Simplify the equation: \[ \frac{5}{2}x - 9 = 9 \]

3. Add 9 to both sides: \[ \frac{5}{2}x = 18 \]

4. Multiply both sides by \( \frac{2}{5} \): \[ x = \frac{2 \cdot 18}{5} = \frac{36}{5} = 7.2 \]

Now, we need to figure out the restrictions based on the original fraction. The expression \( \frac{5}{2}x - 8 \) in the denominator means that \( \frac{5}{2}x - 8 \) cannot equal zero, which leads us to the exclusion: \[ \frac{5}{2}x - 8 \neq 0 \] \[ \frac{5}{2}x \neq 8 \] \[ x \neq \frac{8 \cdot 2}{5} = \frac{16}{5} = 3.2 \]

Finally, combining this with the value we found for \( x \), we can rewrite the original equation as:

\[ \frac{5}{2}x - 9 = 9 \quad \text{and} \quad x \neq \frac{16}{5} \]

So, the correct choice is:

None of the given options are correct. The correct answer would be:

\[ \frac{5}{2}x - 9 = 9 \quad \text{and} \quad x \neq \frac{16}{5} \]