SOLUTION: Find the set of values of x for which the following inequality is true 2x-3/x-5<3/2, x can not equal 5. Please help..

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Question 825159: Find the set of values of x for which the following inequality is true 2x-3/x-5<3/2, x can not equal 5. Please help..
Answer by math-vortex(648) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, there--

Problem:
Find the set of values of x for which the following inequality is true %282x-3%29%2F%28x-5%29%3C3%2F2,  with x%3C%3E5.

Solution:
We have the inequality, %282x-3%29%2F%28x-5%29%3C3%2F2,  with x%3C%3E5. We have the stipulation that 
x%3C%3E5 to avoid division by zero.

Let's work with the related equality, %282x-3%29%2F%28x-5%29=3%2F2. Then we'll use algebra to simplify 
it. Multiply both sides by x-5.
%28x-5%29%2A%28%282x-3%29%2F%28x-5%29%29=%283%2F2%29%2A%28x-5%29

On the left hand side the factor (x-5) in the numerator and the denominator cancel out 
because %28x-5%29%2F%28x-5%29=1. Now we have
2x-3=%283%2F2%29%2A%28x-5%29

Apply the distributive property to remove the parentheses on the right hand side.
2x-3=%283x-15%29%2F2

Multiply both sides by 2 to clear the fraction on the right side.
2%2A%282x-3%29=2%2A%28%283x-15%29%2F2%29

The factor 2 in the numerator and denominator on the right side cancel out because 
2%2F2=1. Apply the distributive property to the left side to clear the parentheses. Now we 
have
4x-6=3x-15

Subtract 3x from both sides. Simplify (4x-3x=x and 3x-3x=0).
4x-6-3x=3x-15-3x
x-6=-15

Add 6 to both sides of the equation to isolate x on the left. Simplify (-6+6=0 and -15+6=-9).
x-6%2B6=-15%2B6
x=-9

The equation x=-9 is equivalent to the original equation, but much easier to understand. It 
represents the critical number where the original inequality shifts from true to false. 


Now we will test the INEQUALITY with a sample number from either side of -9, one greater 
than -9, the other less than -9. We could choose any numbers; we try to choose easy ones. 
Let's start with 0 which is greater than -9.

Let x=0 

%282x-3%29%2F%28x-5%29%3C3%2F2
%282%280%29-3%29%2F%28%280%29-5%29%3C3%2F2
%28-3%29%2F%28-5%29%3C3%2F2
3%2F5%3C3%2F2

x = 0 makes the inequality TRUE. This tells us that all values of x such that x is greater the -9 make the inequality true.

Now we test a value less than -9, say x=-10

Let x=0 

%282x-3%29%2F%28x-5%29%3C3%2F2
%282%28-10%29-3%29%2F%28%28-10%29-5%29%3C3%2F2
%28-23%29%2F%28-15%29%3C3%2F2
23%2F15%3C3%2F2

We need common denominators to evaluate this inequality. Make the denominators be 30.
46%2F30%3C45%2F30

We see that this inequality is FALSE. This tells us that all values of x such that x<-9 make the inequality false.

What about -9 itself? Let's try it.

Let x=-9 

%282x-3%29%2F%28x-5%29%3C3%2F2
%282%28-9%29-3%29%2F%28%28-9%29-5%29%3C3%2F2
%28-21%29%2F%28-14%29%3C3%2F2

Reduce the fraction on the left to lowest terms.
3%2F2%3C3%2F2

When x=-9 both sides of the inequality are equal; -9 is not a solution to the inequality because 3/2 is not less than 3/2.

That's it. The set of values that makes your inequality true is all values of x such that x<-9.

Hope this helps!
Mrs. Figgy