Write an inequality to describe the following and then solve it: the quotient of a number and nine is more than two.
We need to take this sentence and write it as a mathematical inequality. To do that, we’ll first need to know what quotient means. The quotient is the result of a division problem. If you take a dividend and divide it by a divisor, the result is the quotient. If we have the quotient of a number and nine, we’ll let our unknown value be 𝑥. The quotient of a number and nine will be equal to that number divided by nine, 𝑥 divided by nine. And that value is more than two.
If it’s more than two, it’s greater than two. So, we use our greater than symbol. And we’re saying that 𝑥 divided by nine is greater than two, which we could also write as 𝑥 over nine is greater than two. In addition to writing this inequality, we need to solve for the values of 𝑥 that make this inequality true. To get 𝑥 by itself, we’ll multiply both sides of the inequality by nine. 𝑥 over nine times nine equals 𝑥. And two times nine equals 18.
But what sign do we use? Well, if 𝑎 is greater than 𝑏, and 𝑐 is greater than zero. If 𝑐 is positive, then 𝑎 times 𝑐 will still be greater than 𝑏 times 𝑐. Because we’ve multiplied both sides of this equation by a positive value, the inequality sign doesn’t change. And so, the 𝑥-values that make this inequality true are 𝑥-values that are greater than 18. And so, we have 𝑥 over nine is greater than two and 𝑥 is greater than 18.