Solving this equation of absolute value (with the step, please) :1. |x-3|+|x-2|+|x+1|<7 2. |x-3|=x-3 3. show it that |x|≤2 -->
1. You divide the range up into ranges. x>=3, x>=2 x>=-1 and x<=3.
in x>=3 all equations are positive so the equation becomes x-3+x-2+x+1=3x-4
3x-4<7 when 3x<11 or x<11/3 so the equation is true for x<11/3.
in 2<=x<=3 in this interval x-3 is negative so its absolute value can be calculated by 3-x. x-2 and x+1 are positive in this range so the equation becomes
3-x+x-2+x+1=x+2<7 x+2<7 when x<5, so the equation is true in this range and our solution now becomes 2<=x<11/3.
in -1<=x<=2 x-2 is negative and can be replaced with 2-x, x-3 is still negative and can be replaced with 3-x, but x+1 is still positive so our equation becomes
3-x+2-x+x+1=-x+6<7 when -x<1 or x>-1 so our range is now -1<x<11/3. Note that x=-1 the equation is equal to 7 so -1 is not in our solution.
For x<-1 x-3, x-2 and x+1 are negative and can be replaced with their opposites, so the equation becomes 3-x+2-x+-1-x=-3x+4 and -3x+4<7 when
-3x<3 or x>-1 so the equation is never <7 in the interval x<=-1. So we have our solution.
-1<x<11/7 is the solution to |x-3|+|x-2|+|x+1|<7
2. |x-3|=x-3 since if x<3, x-3 is negative, and you cannot have an absolute value that is negative, so this is not true when x<3 (check a value out like 0, |0-3|=0-3 ==> |-3|=-3 ==> 3=-3 which is false. You can also see that if x>=3, |x-3| = x-3 by definition. so this equation is true when x>=3.
3. |x|<=2 --> |(2x^2+3x+2)/(x^2+2)|<=8
First look at (x^2+2) this is always >= 0, since x^2 is always >=0.
Now look at 2x^2+3x+2. This is a parabola, where the vertex is at x = -b/2a = -3/(2(2)) = -3/4. The y value of this parabola is (-3/4)^2 + 3(-3/4) + 2 = 9/16+ -9/4 + 2 = 9/16 + -36/16 + 32/16 = 5/16. so this equation's minimum is positive. So since both (2x^2+3x+2) and (x^2+2) are always positive then (2x^2+3x+2)/(x^2+2) is always positive and we can remove the absolute value signs around it. (2x^2+3x+2)/(x^2+2) <=8. so we want to solve when
Note that we can multiply both sides by (x^2+2) without worring about the <= because (x^2+2) is always positive.
(2x^2+3x+2)<= 8(x^2+2) we get
2x^2+3x+2 <= 8x^2+16
-6x^2 + 3x - 14 <= 0 We can find the maximum of this equation by the same method, find the vertex x = -b/2a = -3/(2(-6)) = 3/12 = 1/4. We have
-6(1/4) + 3(1/4) - 14 = -6/4 + 3/4 - 56/4 = -59/4. So since the maximum is < 0 this inequality -6x^2 + 3x - 14 is always true. So for any value of x, |(2x^2+3x+2)/(x^2+2)| <= 8.