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
