Question 1178838: 2 (| 4-3x |) -3 (| 2x+1 |) < 7
I'm studying for a test and trying to solve this. I always reach the answer 4/3 > x >-1/6 but according to my teacher, the correct answer is x> -1/6 and I keep on going wrong somewhere but can't find where. Help is very much appreciated. Thanks!
Found 3 solutions by MathLover1, greenestamps, ikleyn:
Answer by MathLover1(20850)   (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
We can only guess what you are doing wrong, because you only show us what your wrong answer is, without showing HOW you got your answer.
The two critical points for the expression are where the expressions within the absolute value symbols are equal to zero -- at 4/3 and -1/2. So to solve the inequality you need to examine three cases: (1) x less than-1/2; (2) x between -1/2 and 4/3; and (3) x greater than 4/3.
Solving the first case leads to the equation 11<7, which is never true. So there are no solutions for x<-1/2.
Solving the second case leads to your answer of [-1/6,4/3).
Perhaps you stopped there in your work....
Solving the third case leads to the equation -11<7, which is always true. So all values greater than or equal to 4/3 are also solutions.
That gives what your teacher says is the correct answer.
Answer by ikleyn(52792)  (Show Source):
You can put this solution on YOUR website! .
To help you, I will first show you the PLOT of the function
y = 2*(| 4-3x |) -3*(| 2x+1 |)
Plot y = 2*(| 4-3x |) -3*(| 2x+1 |) (red line) and y = 7 (green line)
From this plot, you can see clearly the solution set to given inequality.
It is the set of points of x-axis, where the red line is BELOW the green line.
Now to the solution of the problem.
There are 2 (two, TWO) critical points, where the linear functions change their behavior.
These points are x =  and x =  , where linear functions under the absolute sign brackets become zero.
These points divide the entire number line in 3 intervals: two infinite and one finite
a) -oo < x <= ; b) < x < and c) <= x < oo.
You need represent your nonlinear function as a linear function in each of these intervals.
You will get 3 linear functions: one for each of these interval.
Then you need to analyze the inequality y < 7 in EACH of these intervals.
It is quite boring procedure, and it requires ACCURACY at each and every step. ---- BUT it is THE ONLY WAY to solve the problem.
One thing may F A C I L I T A T E your analysis : it is the fact that
in intervals a) and c) the linear functions become, ACTUALLY, C O N S T A N T.
Your test is not simple --- it is TRUE.
At which school, college, university, thinking center did you get this assignment ?
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See the lessons
- How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 1
- How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 2
- How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 3
- HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 1
- HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 2
- HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 3
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