Solve the equation three 𝑥 minus two five 𝑥 plus two seven 𝑥 minus three equals zero.
Here, we have three expressions multiplied together to give zero. So, we need to find a value of 𝑥 that will make the product of these expressions equal to zero. To do this, we’re going to use the fact that if we have two values 𝑎 and 𝑏 and their product is equal to zero, then 𝑎 equals zero or 𝑏 equals zero, or 𝑎 and 𝑏 equals zero. Looking at our equation then, it means that three 𝑥 minus two equals zero, or five 𝑥 plus two equals zero, or seven 𝑥 minus three equals zero.
Looking at three 𝑥 minus two equals zero, we can rearrange this by adding two to both sides to tell us that three 𝑥 equals two. To find 𝑥, we would then divide both sides of our equation by three. So, one solution to our equation would be 𝑥 equals two-thirds. When five 𝑥 plus two equals zero, then five 𝑥 would be equal to negative two. And so, 𝑥 would be equal to negative two-fifths. To find our final solution, when seven 𝑥 minus three would be equal to zero, then seven 𝑥 equals three. Therefore, 𝑥 equals three-sevenths. So, our final answer to the solution of the equation is 𝑥 equals two-thirds, 𝑥 equals negative two-fifths, and 𝑥 equals three-sevenths.