Video Transcript
Which of the following values of 𝑦 satisfies the inequality three multiplied by 𝑦 plus four minus 12 is less than 20 minus 17 minus 𝑦? (A) One, (B) two, (C) three, (D) four, or (E) five.
One approach to answering this question would be to substitute each of the possible values of 𝑦 into the inequality and see which results in a true statement. However, this will likely be a time-consuming process. So instead, let’s solve the inequality and then see which of the given options lies in its solution set.
We’ll begin by distributing the three over the parentheses on the left-hand side and the negative sign, which represents a factor of negative one, over the parentheses on the right-hand side. Doing so gives three 𝑦 plus 12 minus 12 is less than 20 minus 17 plus 𝑦. We can then simplify the constants on each side of the inequality to give three 𝑦 is less than three plus 𝑦. Notice that on the left-hand side the constants have canceled one another out completely.
Next, we collect the 𝑦-terms on the same side of the inequality by subtracting 𝑦 from each side to give the equivalent inequality two 𝑦 is less than three. Finally we divide both sides of the inequality by two to give 𝑦 is less than three over two. Remember, if we divide, or indeed multiply, both sides of an inequality by a negative value, we must remember to reverse the direction of the inequality. However, this doesn’t apply here as the value we’re dividing by is positive.
We’ve now solved the given inequality. And we need to determine which of the five given integers lies in the solution set. It may help to represent the solution on a number line. As 𝑦 must be less than three over two, or 1.5, all real numbers that are strictly to the left of this value are in the solution set. It’s clear that the only one of the five integers that lies in this region is one.
So, the answer is (A). The only integer from one to five inclusive that satisfies the given inequality is one.
