Solving Absolute Value Inequalities

Absolute value inequalities can sometimes be challenging to solve, but with the right approach and understanding of the concept, it becomes a manageable task. In this article, we will delve into the world of absolute value inequalities and explore different methods to solve them.

To begin with, let’s clarify what an absolute value inequality is. An absolute value inequality is an inequality that involves absolute value expressions. The absolute value of a number is its distance from zero on the number line, and it is always positive. For example, the absolute value of -5 is 5, while the absolute value of 3 is 3.

Now, let’s discuss how to solve these inequalities step by step. The first step is to isolate the absolute value expression on one side of the inequality. This is done by moving any constants or terms that are not involved within the absolute value away from it. For instance, consider the inequality |2x – 3| < 5. We first move the constant 5 to the other side to obtain |2x - 3| - 5 < 0. After isolating the absolute value, we need to break it down into two separate inequalities. The first inequality is obtained by changing the absolute value sign to a positive sign, while the second inequality is obtained by changing it to a negative sign. In our example, we will have two inequalities: 2x - 3 - 5 < 0 and -(2x - 3) - 5 < 0. The third step is to solve each of the two inequalities separately. In our case, solving the first inequality, we get 2x - 8 < 0, which leads to x < 4. For the second inequality, we have -2x + 3 - 5 < 0, giving us -2x - 2 < 0. By adding 2 to both sides, we find -2x < 2. Dividing both sides by -2 requires us to reverse the direction of the inequality, leading to x > -1.

Once we have obtained the solutions for both inequalities, we need to find their intersection. The intersection of the solution sets will give us the final solution to the original absolute value inequality. In our example, the solutions were x < 4 and x > -1. When we consider the intersection of these solution sets, we find that the final solution is -1 < x < 4. It's important to note that when working with greater than or greater than or equal to inequalities, the final solution is expressed with the symbol 'or' between the two inequalities. For instance, if the final solutions were x > 2 and x ≥ 5, the solution would be x > 2 or x ≥ 5.

In conclusion, solving absolute value inequalities might seem intimidating at first, but by following a systematic approach, the task becomes much more manageable. Remember to isolate the absolute value expression, split it into two separate inequalities, solve each inequality separately, and finally find the intersection of the solutions. With practice, solving absolute value inequalities will become second nature, and you’ll be well-equipped to tackle more complex mathematical problems.