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## Algebra (all content)

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

Lesson 3: Solving absolute value inequalities

# Solving absolute value inequalities: fractions

Sal solves the inequality |2r-3 1/4| < 2 1/2. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• I have trouble with solving combined absolute value inequalities. For example how would you solve and graph: 9< |4x-5| <23
• One way to solve this problem would be to start by rewriting is as 9 < |4x - 5| and |4x - 5| < 23. Then solve each part.

9 < |4x - 5|
|4x - 5| > 9
4x - 5 > 9 or 4x - 5 < -9
4x > 14 or 4x < -4
x > 7/2 or x < -1

|4x - 5| < 23
4x - 5 < 23 and 4x - 5 > -23
4x < 28 and 4x > -18
x < 7 and x > -9/2
-9/2 < x < 7

Finally, the solution is the numbers that satisfy the criteria between -9/2 and 7 for which x > 7/2 or x < -1. So the answer is -9/2 < x < -1 or 7/2 < x < 7. On a number line, this would be represented by shading the numbers between -9/2 and -1 and the numbers between 7/2 and 7.
• i have a really confusing problem related to this and i need some help on it!:
2|4x+1|-5≤ -1
• Okay you want the absolute value by itself. Add 5 to both sides to get 2|4x+1| ≤ 4. Next you divide both sides by 2 to get |4x+1| ≤ 2. Ok now you put 4x+1 ≤ 2 AND 4x+1 ≥ -2. The first one (4x+1 ≤ 2) is simplified to x ≤ 1/4. The second one (4x+1 ≥ -2 ) works out to x ≥ -3/4. Thus, the answer is -3/4 ≤ x ≤ 1/4.
• how so you know if its a "or" or an "and"?
• Think GreaTOR and Less thAND
• How is the absolute value used in the real world?
• ok I think you understand David that - is like debt and + is making money
• how do you know when to put "and" or "or"?
• If the absolute value quantity is less than the other value, use "and". If the absolute value quantity is greater than the other value use "or".
• I have done this before and for example 5647= isn't it the same number?
• I don't know double check it ! ;-)
• When you say, "we'll take the absolute value of it." what do you mean? Are you changing anything such as adding, subtracting, dividing, ect?
• You are simply making the value of the integer positive. It is helpful when considering time and distance, because it is difficult to have negative values for distance and time. That is, unless you're a timelord.
• The absolute value of an expression less than some other number is an AND problem by definition?

So |x|<12 is an AND problem and |x|>12 would be an OR problem? You can tell that by the sign?