5 | x + 6 | + 3 > 5

Question

is the answer form( a <x <b) or (x< a or x>b)

a=

b=

Answer 1

So, these used to confuse me, too.

Remember that absolute value is the distance from zero. Since we only measure distance using positive numbers, absolute value is always positive. Negative seven and seven are both seven away from 0, so |7| = |-7| = 7.

Now, when we solve an absolute value equation, what's inside the absolute value can be positive or negative since both result in the same positive answer.

Throw in inequalities and we have another layer. If the absolute value (the distance from zero) is, let's say, greater than 8. That means that what's INSIDE can be greater than 8 orrrrrr less than -8 (the absolute value of -7 is 7, which is not greater than 8).

If the absolute value is less than 8, that means what's inside has to be less than 8 or greater than -8, aka -8 > x < 8 (again, -9 has an absolute value of 9, which is not less than 8.

Let's look at two sample problems.

|x - 7| > 8

x - 7 > 8

x > 15

x - 7 < -8

x < -1

No number is both less than -1 AND greater than 15, so x > 15 or x < -1.

|x + 5| < 10

x + 5 < 10

x < 5

x + 5 > -10

x > -5

Well, it's certainly possible for a number to be between -5 and 5, so -5 < x < 5.

Hope this helps!

Answer 2

Hi, Summer!

Great question! So before you decide whether it is an AND or an OR inequality, make sure you have isolated the absolute value quality on the LEFT side. Once you have done that, remember LA/GO. Less than AND/ Greater than OR.

AND : a < x < b

OR : x > b OR x < a

I hope this helps! Have a great day!

Mrs S