Which type of absolute value inequality requires the use of a compound inequality to reach the solution?

Let's approach this problem by considering simple cases. We will consider two kinds of absolute value inequalities, depending on whether the absolute value is less than or greater than a constant.



Let's consider this first case where the absolute value of x should be greater than or equal to 2. Intuition would tell us that if x is positive, it should be greater than or equal to 2 while if x is negative it should be less than or equal to -2. These two solutions cannot be combined together. This will be true for every expression with a greater than symbol favoring the absolute value. Answers for this case will just be written as

or

Now let's consider the following case:



For this expression, our x needs to be less than or equal to 2 if it's positive and greater than or equal to -2 if it's negative. We'll notice that these two conditions actually intersect! It is possible to combine the answer into a compound inequality. A compound inequality is an expression where two statements are joined by AND instead of OR (like what happened in the previous case). For this case we'll have the following



ANSWER: Inequalities where 'greater than' symbols favor the CONSTANT will make use of compound inequalities.