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Find all x that satisfy the inequality (2x+10)(x+3)<(3x+9)(x-8) . Express your answer in interval notation.

 Jul 30, 2022
 #1
avatar+302 
+2

Answer: \(x\in(-\infty, -3) \cup (34, \infty)\)

Solution:

Set the equations equal to find critical points:

(2x+10)(x+3)=3(x+3)(x-8)

x=-3 is a critical point, as is x=34.

If this inequality were to be graphed on a number line, it would result in a striped pattern. You can test the middle region (I will test -3

Testing for 0:

(0+10)(0+3)=3(0+3)(0-8)

This region is not shaded.

The regions around it (x<-3 and x>34) are.

Therefore, the answer in interval notation is \(x\in(-\infty, -3) \cup (34, \infty)\).

 Jul 30, 2022
 #2
avatar+124676 
+1

Expanding we have

 

2x^2 + 16x + 30 < 3x^2 -15x - 72       rearrange as

 

x^2 - 31x  - 102 >  0     factor as

 

( x - 34) ( x + 3)  >   0

 

This will be true   when  x < -3   and x > 34

 

So   the intervals that make this true are

 

(  -inf, -3)  U  ( 34, inf)

 

cool cool cool

 Jul 30, 2022

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