Find the solution set of the modulus of 𝑥 plus three equals the modulus of two 𝑥 minus six.
Before we look at this specific question, it is worth doing a quick recap of what the modulus function means. If we look at the example modulus of 𝑥 equals four, then the answer could be positive four or negative four. Therefore, the solution set would contain the two values negative four and positive four.
In order to find the solution set of modulus 𝑥 plus three equals modulus two 𝑥 minus six, we’ll need to solve two equations: firstly 𝑥 plus three equals two 𝑥 minus six and secondly 𝑥 plus three equals the negative of two 𝑥 minus six in the parentheses.
Let’s consider the left-hand equation first. In order to balance this equation, we can firstly add six to both sides. This leaves us with 𝑥 plus nine equals two 𝑥. Our second step would be to subtract 𝑥 from both sides, leaving us with nine on the left-hand side and 𝑥 on the right-hand side. Therefore, one solution is 𝑥 equals nine.
Now let’s look at the right-hand equation. The first thing we will do here is remove the parentheses from the right-hand side. This leaves us with 𝑥 plus three equals negative two 𝑥 plus six. We then begin to balance the equation. Firstly, we have subtracted three from both sides. This leaves us with 𝑥 equals negative two 𝑥 plus three.
Our second step is to add two 𝑥 to both sides, leaving us three 𝑥 equals three. Finally, we divide both sides by three, giving us another solution 𝑥 equals one. Solving these two equations gives us a solution set of one and nine. When we solve the equation, modulus 𝑥 plus three equals the modulus of two 𝑥 minus six.