How do you Solve Absolute Value Inequality?

Solving absolute value equations is similar to solving equations of absolute value, however, there are some additional things to remember. It’s helpful to be comfortable in solving absolute value equations. However, it’s not a problem if you’re working on these in tandem!

Definition of Absolute Value Inequality

The first thing to note is that first of all, an absolute value inequality is an inequality that has an expression that is absolute. For instance,


It is an absolute value inequality since it contains an inequality sign > along with an absolute expression 5 + 5+.

How to Solve an Absolute Value Inequality

The steps to solve an absolute value inequities are similar to the steps needed to solve an equation for absolute value:

1. Separate the absolute value expression from the other end of the equality.

Step 2. Find”positive “version” of the inequality.

Third step: Find the positive “version” of the inequality by multiplying the number on the opposite edge of the inequality with -1 before flipping the sign of inequality.

This is a lot to consider in one go and here’s an example to guide you through the steps.

Find the solution to the inequality the number x:


To calculate this, you need to find 5 + 5 5 by itself on the left-hand right side of this inequality. All you need to add is 3 on each side:

| 5 + 5x | – 3 + 3 > 2 + 3
| 5 + 5x | > 5

Now , there are 2 “versions” of the inequality that we have to resolve two ways to solve it: the negative “version” and the negative “version.”

In this section we’ll assume that the facts are the way they appear: 5+5 + x is greater than 5.


This is a very simple inequalitythat you need to find the variable x in the same way as you would normally. Add 5 to both sides and and then divide each side by 5.

5+5x-5>5-5(subtract five for each side)
5x(/5)>0(/5)(divide each side with five)

This isn’t bad! Another alternative to solving our inequality is the value x is greater than zero. Since there are some absolute values in play It’s time to think about a different alternative.

To comprehend this part it is helpful to know the meaning of absolute value. Absolute value determines the distance of a number from zero. The distance is always positive. meaning that 9.9 is almost nine unit from zero. However, 9 equals nine units from zero.

So |9| = 9, but also |-9| = 9.

you can also find the absolute value of any number wit the help of online absolute value calculator.

Back to the issue earlier. The above work showed that |5 + 5x| > 5. In another way, the total value for “something” is greater than five. The problem is that any number greater than five will be further far from zero that five. The first possibility would be the possibility that “something,” |5 + 5x|, is greater than 5.

This is:


This is the scenario that was discussed previously in the second step.

Think about it a bit more. What else is 5 units from zero? Well, negative five is. Anything that is further from negative five is likely to be further from zero. Thus, the “something” could be a negative number that is further to zero than negative 5. This means that it’s an even more distinctive number, however in terms of technology, it’s smaller than negative five since it’s moving in a negative direction on the line of numbers.

Therefore, the number of our “something,” 5 + 5x, might be lower than -5.


The simplest way to perform math is multiplying the amount on the opposite edge of the inequality five, by negative Then turn the inequality sign upside down:



5+5x-5<-5-5(subtract five from each side)

The two answers to this inequality would be that x>0 or x + 2. Try plugging into a couple of possible ways to verify that the inequality is still true.

Absolute Value Inequalities With No Solution

There’s a situation where there could be no solutions to an absolute value inequities. Since absolute values always have a positive value they cannot be greater or lesser than negative numbers.

Therefore, the equation x | 2 is an unsolved problem since the result for an absolute value equation must the potential to result in positive.

Interval Notation

In order to write out the answer for our main instance using in interval notation consider what the solution appears on the numbers line. The solution we came up with consisted of the following: x > zero or x + 2. On a number line it’s an open dot starting at 0 and a line stretching to positive infinity as well as the open dots at the number -2 with a line stretching away towards negative infinity. These solutions draw attention towards each other but not towards each other, therefore, consider each piece individually.

If x is greater than 0 on a line of numbers There’s an open dot at zero and the line extends out into infinity. In interval notation the open dot is shown with parentheses ( ) as well as closed dots or inequalities containing greater than or equal to or =, will use brackets, [ ]. In other words, for the case of x > 0 then put (0,).

Other half that is x < 2. On the number line, has an unclosed dot starting at the -2 mark and an arrow that extends until the letter –. In interval notation, it’s (-+ 2).

“Or” in interval notation is the union symbol .

Therefore, the solution in interval notation can be described as: