"Solving Absolute Value Inequalities: A Step-by-Step Guide"

Have you ever come across an absolute value inequality and wondered how to solve it? Don't worry, you're not alone! Absolute value inequalities can seem intimidating at first, but with a little bit of practice and understanding, you'll be able to solve them with ease. In this blog post, I'll walk you through the steps of solving an absolute value inequality and provide some exercises for you to practice on your own.

First, let's review what an absolute value is. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of -3 is 3 because -3 is 3 units away from zero on the number line. The absolute value of a number is always positive or zero.

Now that we understand what an absolute value is, let's look at how to solve an absolute value inequality. An absolute value inequality is an inequality that contains an absolute value expression. For example: |x| < 4.

To solve an absolute value inequality, we need to split it into two separate inequalities. This is because the absolute value of a number can be either positive or negative. So, in our example, we would split |x| < 4 into two inequalities: x < 4 and -x < 4.

Next, we need to solve each inequality separately. In our example, the first inequality (x < 4) is already solved. The second inequality (-x < 4) can be solved by dividing both sides by -1 to get x > -4. Remember that when you divide or multiply both sides of an inequality by a negative number, you need to flip the direction of the inequality symbol.

So now we have two solutions to our original absolute value inequality:
x < 4 and x > -4.
To write our final solution, we combine these two inequalities using the word "and" to get: x > -4 and x < 4.
This means that any value of x that is greater than -4 and less than 4 will make the original absolute value inequality true.


Let's check our solution by plugging in some values for x. If we plug in x = -5, we get |-5| < 4 which simplifies to 5 < 4. This is not true, so x = -5 is not a solution to our original absolute value inequality. If we plug in x = 0, we get |0| < 4 which simplifies to 0 < 4. This is true, so x = 0 is a solution to our original absolute value inequality.

Exercise for Practice:
Now it's your turn to practice solving an absolute value inequality! Try solving the following absolute value inequality on your own: |2x + 3| > 5.


Solving absolute value inequalities may seem challenging at first, but with a little bit of practice and understanding, you'll be able to solve them with ease.
Remember to split the original absolute value inequality into two separate inequalities and solve each one separately. Then combine your solutions using the word "and" to get your final answer. Keep practicing and you'll be a pro at solving absolute value inequalities in no time!