Solving Inequalities: 0<(2x+1)(x-2) or 0>(2x+1)(x-2)?

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In summary: So x=2 is not a solution2 is in the interval (a,b)-2 is in the interval (a,b)In summary, the conversation discusses solving the inequality (4x-4)/(x+2)<2x-3 by factoring and determining the sign of the product (2x+1)(x-2). The expert suggests considering intervals and checking one point in each to determine if the inequality is satisfied. The final solution is x>2, with the reminder to also consider the interval -2<x<-1/2.
  • #1
deryk
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0
Im trying to solve (4x-4)/(x+2)< 2x-3

I get it down to 0<(2x+1)(x-2) if x+2>0

0>(2x+1)(x-2) if x+2<0

Are these right so far? I am not sure what to do now with the product being bigger or smaller than 0. Thanks for your time.
 
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  • #2
deryk said:
Im trying to solve (4x-4)/(x+2)< 2x-3

I get it down to 0<(2x+1)(x-2) if x+2>0

0>(2x+1)(x-2) if x+2<0

Are these right so far? I am not sure what to do now with the product being bigger or smaller than 0. Thanks for your time.
Right so far.
You have products
the sign of a product depends on the sighns of the factors
ab>0
means
a>0 and b>0
or
a>0 and b<0
ab<0
means
a>0 and b<0
or
a<0 and b>0

another way to think about it is (2x+1)(x-2) is a continuos function
find out where the zeros are
call them a and b with a<b
consider the intervals (since -2 is also an important number)
x<-2
-2<x<a
a<x<b
b<x
all the points in one of these intervals satisfy the inequality or none do
so checking one point in each intervals tells you if the whole interval satifies the inequality
 
Last edited:
  • #3
thanks lurf lurf .I got x>2. Does anyone know if that's right?
 
  • #4
deryk said:
thanks lurf lurf .I got x>2. Does anyone know if that's right?
you missed -2<x<-1/2
consider for example x=-1
(4x-4)/(x+2)< 2x-3
(4(-1)-4)/((-1)+2)< 2(-1)-3
(-4-4)/1<-2-3
-8<-5
 

1. What does it mean to solve an inequality?

Solving an inequality means finding all the possible values of the variable that make the inequality true.

2. How do you solve an inequality with a compound expression?

To solve an inequality with a compound expression, you need to break it down into simpler inequalities and solve each one separately. Then, you can combine the solutions to get the final solution.

3. How do you know if you need to flip the inequality sign when solving?

You need to flip the inequality sign when multiplying or dividing both sides of the inequality by a negative number. This is because multiplying or dividing by a negative number will change the direction of the inequality.

4. Can you solve an inequality with variables on both sides?

Yes, you can solve an inequality with variables on both sides by first simplifying the expression and then isolating the variable on one side of the inequality sign.

5. How can you check if your solution to an inequality is correct?

You can check if your solution is correct by substituting the value of the variable into the original inequality and seeing if it makes the inequality true. Additionally, you can graph the inequality and see if the solution falls within the shaded region.

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