Solve the following inequation :
- 3 (x -7) ≥ 15 - 7x > x + 1/3 ; x ε R
Answers
Answer:
X3-11/4X2+5/4X+1/2=0
The highest term of this equation is to the third power telling us there will be three solutions. The first thing to do is get rid of the fractions so it's easier to work with, this won't change the solutions. Multiply the entire equation, both sides, by 4.
(4)*(X3-11/4X2+5/4X+1/2)=(4)*0
4X3-11X2+5X+2=0
This cannot be factored by grouping. Therefore we will have to factor using the free term. The free term is 2. This is the number that has no X variable associated with it. First find the factors of the free term. These are : 2, 1.
We must find which one is a solution for the equation. We do this by setting X equal to the factor until we reach an equation of 0=0 indicating it is a true statement and thus a solution.
I will plug in 1.
4(1)3-11(1)2+5(1)+2 =0
4(1) - 11(1) +5(1) +2=0
11-11=0
0=0
this is true therefore one is a solution or x=1.
We can rearrange x=1 to find one of the factors of this equation
X-1=1-1
X-1=0
(X-1) is a factor of the given equation.
Next we must rearrange the equation so that (x-1) can be factored out of the terms.
4X3-11X2+5X+2=0
Because we want the factor to be X-1 i will reorganize the terms so each grouping will contain the same coefficient.
-11X2=-4X2-7X ----> (4X3-4X3)-7X2+5X+2=0
Now i will factor 4X2 from the first two terms.
4X2(X-1)-7x2+5X+2=0
5x=7x-2x ----> 4x2(X-1)-(7x2+7x)-2x+2=0
Now i will factor -7X from the next pair of terms.
4X2(X-1)-7X(X-1)-2X+2=0
The last two coefficients both contain a 2 so I will factor out 2.
4X2(X-1)-7X(X-1)+2(X-1)=0
Now following the distributive property the equation can be rewritten.
(x-1)(4X2-7X+2)=0
The quadratic can be factored now resulting in three factors of the original equation.
(x-1)(4X+1)(x-2) =0
set each factor equal to 0 to find the solutions.
X-1=0 ---> X=1
4X+1=0 ---> X=-1/4
X-2=0 ----> X=2
in order from the smallest to the largest they are
-1/4,1,2