Solve the following inequality: one-fourth is less than four-ninths 𝑦.
There are two different ways to solve this inequality. First, we need to realize that we’re trying to isolate 𝑦, get it by itself. So to get rid of 𝑦, we need to divide by the four-ninths because four-ninths is being multiplied to 𝑦. So the opposite would be to divide. The four-ninths cancel. And we need to take one-fourth divided by four-ninths. So when dividing by fractions, we actually multiply by the second fraction’s reciprocal.
So instead of dividing by four-ninths, we’re multiplying by nine-fourths. And then, when we multiply fractions, we multiply the numerators, the numbers on top of the fraction. So we have one times nine which is nine. And then we multiply the denominators on the bottom, four times four which is 16. So we have nine 16ths is less than 𝑦.
Another way to solve an inequality like this would be to get rid of the fractions, the numbers on the bottom. Eliminating those would mean that we wouldn’t have fractions anymore. So let’s think about this, the one-fourth.
If we wanted to get rid of a four on the bottom, we would need to multiply by a four on the top. But whatever we do to one side, we must do to the other. So on the left side of the inequality, the fours cancel. And on the right, the fours from the numerators, because you could imagine the four as four over one, so four times four is 16. And nine times one is nine. So now we still have a nine on the bottom, the denominator. So to get rid of that, we multiply both sides of the inequality by nine. So we have that nine is less than 16𝑦. So to resolve for 𝑦, we divide both sides by 16.
So once again, we get that nine 16ths is less than 𝑦.