# How to write absolute value inequalities in interval notation

Minus two and plus four are six units apart.

### Absolute value equations and inequalities worksheet

And you really should visualize a number line when you do this, and you'll never get confused then. They're asking you for the x-values that will make the absolute-value expression greater than a negative number. Since the absolute value will always be greater than any negative number, the solution must be "all x" or "all real numbers". Or you divide both sides of this by 7, you get x is greater than or equal to 3. The something is the piece in the middle, where the variable is. It's going to be less than 12 away from 0. In this inequality, they're asking me to find all the x-values that are less than three units away from zero in either direction, so the solution is going to be the set of all the points that are less than three units away from zero. You subtract 3 from both sides. However, the number 2. Let's do a harder one. If you have something like f of x, the absolute value of f of x is less than, let's say, some number a.

And then we can draw the solution set. Example 1 Solve each of the following.

In interval notation, it would be everything between negative 12 and positive 12, and not including those numbers. Since the absolute value will always be greater than any negative number, the solution must be "all x" or "all real numbers".

## How to solve inequalities with absolute values on both sides

So in order to satisfy this thing in this absolute value sign, it has to be-- so the thing in the absolute value sign, which is 5x plus it has to be greater than negative 7 and it has to be less than 7, in order for its absolute value to be less than 7. And actually, we've solved it, because this is only a one-step equation there. This is an inequality. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero. This process can feel a bit weird, so I'll give a couple examples of how it works. So remember what I told you about the meaning of absolute value. We can draw a number line to represent the condition to be satisfied. The second method is to leave the compound inequality intact and perform solving procedures on the three parts at the same time. That translates to that, which translates to f of x greater than negative a and f of x less than a.

To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently. If you subtract 63 from both sides of this equation, you get 2x is less than negative This is an inequality.

Therefore, in this case there is no solution since it is impossible for an absolute value to be strictly less than zero i. Then, when you take its absolute value, it'll become a plus 5.

## Absolute value inequalities graph

If this thing, this 5x plus 3, evaluates anywhere over here, its absolute value, its distance from 0, will be less than 7. We solve them independently. Instead, we will mostly use the geometric definition of the absolute value: The absolute value of a number measures its distance to the origin on the real number line. Let's subtract 63 from both sides of this equation, and you get 2x-- let's see. Translate into English: we are looking for those real numbers x whose distance from the origin is less than 5 units. Oh, I just realized I made a mistake here. And then we can just solve these. This is an inequality. It's going to be less than 12 away from 0. It means how far away you are from 0.

Because the variable is contained within one interval. You have to be in this range. And I really want you to internalize this visualization here.

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