When you study the graphs of absolute value equations, you can see the three cases of one solution, no solution, and two solutions graphically. Since this means that the distance from zero on the number line is zero, you end up with only one equation. In the case of only one solution, you end up with an absolute value expression equal to zero. So this is why we end up with two different equations. Is the absolute value is 2, then all you know is that \(5x + 1\) is 2 units from zero on the number line. Now think of an equation where the absolute value part is isolated, such as \(|5x + 1| = 2\). This is why the absolute value is always positive – it is representing a distance. You can think of the absolute value of any number as representing how far it is from zero on the number line. You will always follow those two steps when solving any absolute value equation. Notice that in both examples, the steps were the same as before. Use this to determine when there are no solutions to an absolute value equation. The absolute value of any number is always positive. This is not possible so there are no possible x-values that make this equation true. Here, we have the absolute value of something is negative. Why? The absolute value of any number is positive. To isolate the absolute value, subtract 3 from both sides.Īt this step, it can be determined that there are no solutions to the equation. We will look more closely at why this happens, but first let’s look at how you might end up with no solutions. Therefore, we will solve two equations without the absolute value: one where the 13 is positive and one where 13 is negative. ![]() ExampleĪs mentioned, the absolute value part is already isolated. Whether or not this first step applies or not, you will always have zero, one, or two solutions to any absolute value equation. In this first example, the absolute value part of the equation is already isolated, so only step two will apply. ![]() Let’s try these steps out with some examples! Examples ![]() This sounds complicated, but it is only a step or two more than solving the typical linear equation. Then, you will write two equations based on the definition of absolute value (though sometimes, there will end up only being one equation). When given an absolute value equation, you will first need to isolate the absolute value part of the equation. Steps for solving absolute value equations Absolute value equations with no solutions or one solution.Steps for solving absolute value equations.In this lesson, we will look at a few examples to understand how to solve these equations and also take a bit of a look at this idea of distance as it relates to solving absolute value equations. Solving absolute value equations is based on the idea that absolute value represents the distance between a point on the number line and zero.
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