Find the solution set of the inequality, the absolute value of 𝑥 plus four is less than nine.
To solve the absolute value, we’ll need to break this equation down into two pieces. We have the case when 𝑥 plus four is less than nine, the positive case. But we also have when negative 𝑥 plus four is less than nine.
We can start solving the problem on the left by distributing our negative. What we have now is negative 𝑥 minus four is less than nine. And then we treat this problem just like any other inequality. We isolate 𝑥. I can do that by adding four to both sides of the inequality. Negative 𝑥 minus four plus four leaves us with negative 𝑥. And nine plus four equals 13.
But what we have now is a negative 𝑥 and we need positive 𝑥. To do that, we’re going to multiply the entire inequality by negative one. Negative one times negative 𝑥 equals 𝑥. When dealing with inequalities, we multiply negative one times the inequality, and that means we flip the sign, 𝑥 is greater than. Multiplying negative one times 13, 𝑥 is greater than negative 13.
We look at our second inequality, 𝑥 plus four is less than nine. We can subtract four from both sides of the inequality, and we have 𝑥 is less than five.
Here’s what we know, that 𝑥 is greater than negative 13 and 𝑥 is less than five, which means that our 𝑥 falls between negative 13 and five. We can write that in a compound inequality like this: 𝑥 falls between negative 13 and five. Another way to write this would be with brackets facing outward. And it says the same thing; 𝑥 is between negative 13 and five.
All of these three are ways to represent the solution to this inequality. Our answer choice is looking for a solution set like this: 𝑥 falls between negative 13 and five.