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.
1.
|x - 3| < 4 - 2x
2x - 4 < x - 3 < 4 - 2x
Splitting them up, you get
2x - 4 < x - 3 and x - 3 < 4 - 2x
x < 1 and x < 7/3
Hence, the answer is x < 1.
"and" here refers to "intersection", which means the answer is only the part where the 2 separate inequalities "intersect".
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.
Didn't you do the same thing for the first question in which the 'a' term contains x?
Originally posted by Audi: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