Write the set of all solutions to the equation absolute value 𝑥 plus six equals three.
Essentially, we wanna know what can we plug in for 𝑥 and when we take the absolute value of that number plus six is equal to positive three. Well, there are actually two numbers that you can plug into the absolute value that would equal three. The absolute value of positive three is equal to positive three and the absolute value of negative three is equal to positive three. This is because the absolute value would turn any number positive. So this means we should take 𝑥 plus six and set it equal to positive three and also set it equal to negative three.
So we’re taking 𝑥 plus six and setting it equal to positive three and taking 𝑥 plus six and setting it equal to negative three. Let’s work with 𝑥 plus six equals three first. We get 𝑥 equals negative three after subtracting six from both sides of the equation; we need to do the same thing to the other equation. And that results in 𝑥 equals negative nine. So the two solutions to this equation are negative three and negative nine. Let’s go ahead and plug them back in just to be safe.
Looking at the equation on the left first, negative three plus six is equal to three. So we have the absolute value of three equals three. And the absolute value of any number turns a positive, so are already positive three. Is it? Three. So three equals three. Now the other equation negative nine plus six is equal to negative three. However, the absolute value of negative three is positive three. So three equals three. Therefore, the solution set is negative nine and negative three.