Question 828661
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Hi, there--

THE PROBLEM:
Solve the following inequality. Enter the answer in interval notation

|3-4x|<7

A SOLUTION:
The absolute value inequality has two branches. Either the expression inside the absolute 
value signs (3-4x) can be positive or it can be negative. In either case, the absolute value 
returns a positive value. We deal with each case separately.

CASE I :: Suppose 3-4x is positive. Then
3 - 4x < 7

To simplify, we subtract 3 from both sides of the inequality.
-4x < 7-3
-4x < 4

To isolate x on the left, we divide both sides of the inequality by -4. Whenever we divide the 
terms of an inequality by a negative number, we must reverse the direction of the inequality.
x > -1

CASE II :: Suppose that 3 - 4x is negative. Then
-(3 - 4x) < 7

Use the distributive property to clear the parentheses.
-3 + 4x < 7

To simplify, add 3 to both sides of the inequality.
4x < 7 + 3
4x < 10

Divide both sides of the equation by 4.
x < 10/4
x < 2.5

Combine the two cases, we have 
x > -1 OR x < 2.5

The numbers that satisfy this inequality are between -1 and 2.5. In interval notation, we 
write (-1, 2.5).


Hope this helps! Feel free to email if you have any questions about the solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com
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