How do you use the zero-product property to solve \[\left( x-5 \right)\left( 2x+7 \right)\left( 3x-4 \right)=0\]?

Hint: In order to find the solution to the given question, that is to use the zero-product property to solve \[\left( x-5 \right)\left( 2x+7 \right)\left( 3x-4 \right)=0\] apply the concept of the zero-product property that the multiplication of several factors is zero if and only if at least one of the factors is zero. This means you can equate each factor equal to zero and the find the values of variable\[x\]for which this zero-product property will always hold for the expression \[\left( x-5 \right)\left( 2x+7 \right)\left( 3x-4 \right)=0\].

Complete step-by-step answer:
According to the question, given equation in the question is as follows:
\[\left( x-5 \right)\left( 2x+7 \right)\left( 3x-4 \right)=0\]
By the zero-product property, we know that the multiplication of several factors is zero if and only if at least one of the factors is zero. So, your expression is equal to zero if and only if one of the three factors equals to zero.
So, by applying the result of zero-product in the given equation we get:
\[\Rightarrow \left( x-5 \right)=0,\text{ }\left( 2x+7 \right)=0\text{ }\!\!\And\!\!\text{ }\left( 3x-4 \right)=0\]
Hence, we can write that \[\left( x-5 \right)=0\] if and only if \[x=5\].
And \[\left( 2x+7 \right)=0\] if and only if \[x=-\dfrac{7}{2}\].
Finally, \[\left( 3x-4 \right)=0\] if and only if \[x=\dfrac{4}{3}\].
Therefore, we conclude that the solution of the equation \[\left( x-5 \right)\left( 2x+7 \right)\left( 3x-4 \right)=0\] are the following values: \[x=5,\text{ }x=-\dfrac{7}{2}\text{ }\!\!\And\!\!\text{ }x=\dfrac{4}{3}\].

Note: Students make mistakes while calculating the value of the variable from the factors. They generally end up making the mistakes in sign while solving the expressions when factors are equated to zero. It’s important to be careful with the sign of the term and recheck the answer again once solved.