To solve the equation [tex]\((2x - 5)(3x - 1) = 0\)[/tex], we need to use the property that if the product of two factors is zero, then at least one of the factors must be zero. This is known as the zero-product property.
1. First Factor: Solve [tex]\(2x - 5 = 0\)[/tex]
[tex]\[
2x - 5 = 0
\][/tex]
Add 5 to both sides:
[tex]\[
2x = 5
\][/tex]
Divide both sides by 2:
[tex]\[
x = \frac{5}{2}
\][/tex]
So, one solution is [tex]\(x = \frac{5}{2}\)[/tex].
2. Second Factor: Solve [tex]\(3x - 1 = 0\)[/tex]
[tex]\[
3x - 1 = 0
\][/tex]
Add 1 to both sides:
[tex]\[
3x = 1
\][/tex]
Divide both sides by 3:
[tex]\[
x = \frac{1}{3}
\][/tex]
So, another solution is [tex]\(x = \frac{1}{3}\)[/tex].
Therefore, the solutions to the equation [tex]\((2x - 5)(3x - 1) = 0\)[/tex] are [tex]\(x = \frac{5}{2}\)[/tex] and [tex]\(x = \frac{1}{3}\)[/tex].
The correct answer is:
[tex]\[
x = \frac{5}{2} \text{ or } x = \frac{1}{3}
\][/tex]
