Solve the inequality and express the solution in interval form.

Solution:

For convenience we'll compare an appropriate expression to 0 instead of 1. Then we can find x-values of interest to determine where to shade (i.e. where the inequality is true).

 





















This graph shows our solution another way. Notice that we graph both sides of the inequality. Then we can actually see when the left side is greater...

(x-a)/(x+a) - (x+a)/(x-a)=7/12 solve the equation

then...









so x=a/7 or x=-7a but in either case
 and 

Solve |2x|<=|x-3|

We start by finding points from which we will find the boundaries for the intervals 

2x=x-3 or 2x=-x+3
x=-3 or x=1 

Next we test all intervals 

On x<-3 we test x=-4 and see 8<7 (false) 

On -3 On x>1 we test x=2 and see 4<1 (false)

Our solution interval: [-3,1]


:)

In an ordered field if a>b show that the following inequality holds

find k such that for every real x we have (1 + kx)/(1 + x^2) < k

Assuming k is any real number... 
 
leads to
 
then 
 
and combining them we get 
 
after simplifying the numerator... 
 
Notice that  is always positive, so to determine when the sign of the expression will be less than zero we only need to look at the numerator: 
 
using the quadratic formula to find x... 
 
then... 
 
then... 
 
which only has a positive real value when 
 and  

or 
 and  
So we notice x is only real on the interval 
 
This interval gives us boundary conditions to test. After testing (using a graphing utility) we see only the values k>4/3 satisfies the inequality for ALL values of x

Solve the inequality and express the solution in interval form.

Solve the inequality and express the solution in interval form.