Question Video: Finding the Solution Set of a Quadratic Equation Involving Absolute Value Mathematics

Video Transcript

Write the set of all solutions to the equation the absolute value of 𝑥 minus five multiplied by the absolute value of 𝑥 plus nine is equal to zero.

In this question, we need to write down the set of all solutions to a given absolute-value equation. To do this, we first recall that a solution to an equation means that both sides of the equation are balanced. Or another way of saying this is the equation is satisfied. In other words, the absolute value of 𝑥 minus five times the absolute value of 𝑥 plus nine needs to be equal to zero.

And we can immediately see something from this equation. We have a product on the left-hand side of this equation. And this must be equal to zero. And we recall for a product to be equal to zero, one of the factors must be equal to zero. Therefore, for this equation to be true, either the absolute value of 𝑥 minus five needs to be equal to zero or the absolute value of 𝑥 plus nine needs to be equal to zero. And we can solve both of these cases separately. We just need to recall the absolute value of a number tells us the size of the number. In other words, it’s just the value of the number where we ignore the sign.

Therefore, for the absolute value of a number to be equal to zero, that number must be equal to zero. In other words, we just need to solve 𝑥 minus five is equal to zero or 𝑥 plus nine is equal to zero. And now we can just solve both of these separately for 𝑥. In our first equation, we add five to both sides to see that 𝑥 is five. And in our second equation, we subtract nine from both sides to see 𝑥 must be equal to negative nine. And these are all of the possible solutions.

Remember, the question wants us to write this as a set. And this gives us our final answer. We were able to show the set of all solutions to the equation the absolute value of 𝑥 minus five times the absolute value of 𝑥 plus nine is equal to zero is the set containing negative nine and five.