The product of two numbers is less than 6 times the sum of these numbers. The difference between these numbers is 12. Which of the following expresses this?

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Quizplus · Accepted Answer

When doing problems like this, the best way to ensure you do not make any mistakes is to create your own variables and different equations for each piece of information you are given. Then you can combine the equations. Start with the simpler statement that the difference between these numbers is 12. Since we don't know what the two numbers are, we will use variables (x and y) to represent them. x - y = 12 Then, you isolate a variable in one equation. Let's isolate the y. We can start with the subtraction equation. x - y = 12 In order to isolate y, we need to move x from the left hand side of the equation to the right hand side. In order to avoid having our y be negative, we will do this in two steps. First, we will add the y to the right hand side of the equation, making it positive. x - y + y = 12 + y x = 12 + y Now, we subtract the 12 from both sides in order to move it over to the left hand side of the equation and isolate the y. x - 12 = 12 - 12 + y x - 12 = y Therefore, we know that y is equal to x - 12, so we can substitute it into the other equation in place of the y. The second equation should show that the product of the two numbers is less than 6 times the sum of these numbers. xy < 6 (x + y) Now we substitute x - 12 for y in this equation: x (x - 12) < 6 (x + x - 12) Begin to simplify by distributing the x on the left hand side of the inequality. x (x - 12) < 6 (x + x - 12) (x • x) - (x • 12) < 6 (x + x - 12) x² - 12x < 6 (x + x - 12) Distribute the 6 to the values inside of the parentheses on the right hand side of the inequality. x² - 12x < 6 (x + x - 12) x² - 12x < (6 • x) + (6 • x) - (6 • 12) x² - 12x < 6x + 6x - 72 Combine like terms (x) on the right hand side of the equation. x² - 12x < 6x + 6x - 72 x² - 12x < 12x - 72 Combine like terms agin (x) by subtracting the 12x from both sides of the equation. x² - 12x - 12x < 12x - 12x - 72 x² - 24x < - 72 Now this is a situation where you want to look at your answer choices. We can clearly see that the answer choices all have a zero on the right hand side of the inequality. Therefore, we are going to move our 72 over to the lefthand side, making the right hand side 0. x² - 24x + 72 < - 72 + 72 x² - 24x + 72 < 0

The Product of Two Numbers Is Less Than 6 Times