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## Algebra (all content)

### Course: Algebra (all content)>Unit 8

Lesson 3: Solving absolute value inequalities

# Intro to absolute value inequalities

To solve inequalities with absolute values, use a number line to see how far the absolute value is from zero. Split into two cases: when it is positive or negative. Solve each case with algebra. The answer is both cases together, in intervals or words. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

5|8-y|=30

Thanks! • quick question. When he breaks down the inequality from Abs.val. 5x+3 less than seven, why does he reverse the inequality when saying 5x+3>-7? • I think your question seems like this I5x+3I<7
so 5x+3 is always less than 7.
so consider few cases
If 5x+3=2; the above given condition is satisfied.
now if 5x+2 = -8 then abs value of 5x+3 is greater than 7 so inequality fails.
so if you break the above mod value then its value must lie in between (-7,7)
and the inequality looks like -7< 5x+3<7.
• For absolute value inequalities, when the constant on one side is negative, is there then no answer, like absolute value equations, or do you solve it normally? • Great question!
It is a little different with absolute value inequalities. Sounds like you already are aware that with absolute value equations if you have a negative constant on one side after the absolute value is isolated, then the equation has no solution. The reason this happens is the absolute value always creates a positive number. And, a positive number will never = a negative number.

So, let's extend that concept to inequalities.
1) An absolute value < a negative: In this situation, we have a positive value < a negative value. That would always be false. This is a contradiction... therefore, this inequality has "No Solution".
2) An absolute value > a negative: In this situation, we have a positive value > a negative value. That would always be true. It is an identity... the solution to the inequality becomes "all real numbers".

Hope this helps.
• Please help I am so dumbfounded. Do you know how to determine an interval when you have two variables? I have a problem:
"Write your answer as an interval, or as a number if applicable. Find all real number of x and y such that: |7x-iy| = y-i6x "
So I know that you can only add the real terms with the real and the imaginary terms with the imaginary. I tried ignoring the imaginary values since those seem to usually cancel out in some absolute value situations. I was confirmed to be correct that y = [0,inf). However I have been spending way too long trying to figure out what x is I tried a lot of things and I'm just stuck. • |7𝑥 – 𝑦𝑖| = 𝑦 – 6𝑥𝑖
Immediately, I can tell that the only value 𝑥 can be is 0. Why? Well the magnitude of a complex number is always a real number. In this equation, we have equated the magnitude of a complex number to another complex number, and so we must make the imaginary part of the second complex number 0. Hence, 𝑥 = 0. Making this substitution:
|-𝑦𝑖| = 𝑦
√𝑦² = 𝑦
But since magnitude is always nonnegative, we have 𝑦 ∈ [0, ∞). Therefore, the solution is:
𝑥 ∈ {0}
𝑦 ∈ [0, ∞)
Comment if you have questions.
• how would you solve a question like |x| + |x-1| <5 ? • what is the big difference between Equations and Inequalities? he corrects it at . it's always being corrected. • An equation means that it has an equal sign. When this happens, it shows that there is the answer, whereas with inequalities, it says that it is greater than this.

For instance, say you had: |x-4| = 7
This would work out to be: x = 11 OR x = -3

If we made this problem an inequality, it would work out the same but the signs would be different: |x - 4| < 7
x < 11 AND x > -3
• How do you solve -7|2X-12|<70? • -7|2X-12|<70

First, divide both sides of the inequality by -7. Don't forget to switch the sign around for dividing by a negative number.

|2X-12| > -10

And then the answer is all real numbers. Think about it, no matter what X is, after you plug in the numbers, the absolute value sign will make the left hand side be at least 0. It is impossible to get an answer less than 0, let alone -10. So all values of X will provide an answer greater than -10, so all real numbers will work for this inequality.
• pov all of the comments are from 10 years ago • So basically to sum this video up you have to find two statements that satisfy the equation such as:
|x| > 15 (The absolute value of X is 15 away from zero)
|x| < 15 and > -15

right? Or am I missing something?  