Solve the inequalities, giving your answers in exact form.
|x-3| < 4-2x
The lecture notes provided the solution using the means of a
graphic calculator. I want to know what is the correct method if I
want to solve this question analytically instead of using the GC. I
followed the rule where if |x| < a, -a < x < a
So it becomes
-(4-2x) < x - 3 < 4- 2x
Solving, I get x <1 or x < 7/3. If I use GC method, I will
know the answer is x<1. But if I use analytical method, how do I
know I must reject x < 7/3 ?
2. Solve the inequality | 3x -12 | < 5/2x , giving your
answers in exact form. How do I present my working in analytical
method given that there are terms consisting of 'x' on both sides
of the equation? I cant apply the rule |x| < a, -a < x < a
because this rule is applicable only when 'a' does not contain an
'x' term.
Thanks.
For 1), notice that if:
x ≥
3, |x - 3| < 4 - 2x is the same as x - 3 < 4 - 2x
x - 3 < 4 - 2x
3x < 7
x < 7/3
Our condition in the first place is x ≥
3, yet we have x < 7/3 now, which is clearly less than
3.
Thus, we will reject this answer.
Now, if:
x < 3, |x - 3| < 4 - 2x is the same as -(x - 3) < 4 -
2x
-(x - 3) < 4 - 2x
3 - x < 4 - 2x
x < 1
Our condition is x < 3, and we have x < 1. Thus x < 1
is a solution.
For 2), if:
x ≥
4, | 3x -12 |
< 5/2 x is the same as 3x - 12 < 5/2 x
3x - 12 < 5/2 x
1/2 x < 12
x < 24
Our original condition is x ≥
4 and now we are told that x < 24.
Thus, 4 ≤
x < 24 is a solution
On the otherhand, if:
x < 4, | 3x -12 |
< 5/2 x is the same as -(3x - 12) < 5/2 x
-(3x - 12) < 5/2 x
12 - 3x < 5/2 x
24 - 6x < 5x
11x > 24
x > 24/11
Our condition is x < 4, and we are given x > 24/11.
Thus, 24/11 < x < 4 is a solution.
We are not done though.
Notice that our solutions are 4 ≤
x < 24 and 24/11 < x < 4
These can be combined into a single inequality equation, 24/11
< x < 4