To determine if Joelle's proportion [tex]\(\frac{4}{6} = \frac{5}{x}\)[/tex] is correct, we need to solve for [tex]\(x\)[/tex] and check whether the relationship holds true through the necessary mathematical steps.
First, we can use the property of cross-multiplication to check if the given proportion is balanced. Cross-multiplying the two ratios in the proportion, we get:
[tex]\[
4 \cdot x = 6 \cdot 5
\][/tex]
Next, simplify the right-hand side:
[tex]\[
4x = 30
\][/tex]
Then, solve for [tex]\(x\)[/tex] by dividing both sides of the equation by 4:
[tex]\[
x = \frac{30}{4}
\][/tex]
This simplifies to:
[tex]\[
x = 7.5
\][/tex]
Next, we need to verify if substituting [tex]\(x = 7.5\)[/tex] back into the original proportion maintains the equality.
Substitute [tex]\(x = 7.5\)[/tex] into the right side of the proportion:
[tex]\[
\frac{5}{x} = \frac{5}{7.5}
\][/tex]
Now we calculate the right side:
[tex]\[
\frac{5}{7.5} = \frac{5}{15/2} = \frac{5 \cdot 2}{15} = \frac{10}{15} = \frac{2}{3}
\][/tex]
And then we simplify the left side:
[tex]\[
\frac{4}{6} = \frac{2}{3}
\][/tex]
Since both sides of the proportion are equal ([tex]\(\frac{2}{3} = \frac{2}{3}\)[/tex]), we confirm that the proportion holds true. Therefore, Joelle's proportion [tex]\(\frac{4}{6} = \frac{5}{x}\)[/tex] is correct.
Thus, the correct answer is:
A. Joelle's proportion is correct.
