To solve the equation [tex]\((x-8)(5x-6)=0\)[/tex] using the zero-product property, let's go through it step-by-step.
The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero.
So, we set each factor in [tex]\((x-8)(5x-6)=0\)[/tex] to zero and solve for [tex]\(x\)[/tex].
Step 1: Set the first factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ x - 8 = 0 \][/tex]
To solve this, add 8 to both sides:
[tex]\[ x = 8 \][/tex]
So, one solution is:
[tex]\[ x = 8 \][/tex]
Step 2: Set the second factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ 5x - 6 = 0 \][/tex]
To solve this, add 6 to both sides:
[tex]\[ 5x = 6 \][/tex]
Now, divide both sides by 5:
[tex]\[ x = \frac{6}{5} \][/tex]
So, the second solution is:
[tex]\[ x = \frac{6}{5} \][/tex]
We can write [tex]\(\frac{6}{5}\)[/tex] as a mixed number:
[tex]\[ x = 1 \frac{1}{5} \][/tex]
Conclusion:
The solutions to the equation [tex]\((x-8)(5x-6)=0\)[/tex] are:
[tex]\[ x = 8 \][/tex]
and
[tex]\[ x = 1 \frac{1}{5} \][/tex]
Therefore, the correct answer is:
[tex]\[ x = 8 \text{ or } x= 1 \frac{1}{5} \][/tex]
