To solve the equation [tex]\((2x - 5)(3x - 1) = 0\)[/tex], we use the zero-product property, which states that if the product of two expressions is zero, then at least one of the expressions must be zero. Therefore, we can set each factor in the equation to zero and solve for [tex]\(x\)[/tex] individually.
1. First, consider the factor [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. Next, consider the factor [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]. This matches the third option provided:
[tex]\[
x = \frac{5}{2} \text{ or } x = \frac{1}{3}
\][/tex]
