How to Solve Inequalities with Absolute Value and Fractions?

  • Thread starter SengNee
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In summary: You can check that no value of x outside of those ranges will work. That is, it is necessary (though not sufficient) that x lies in the interval (1/6, 3/2) for the solution to work. This can be verified by plugging in any value of x in that interval into the original inequality and seeing that it holds. So yes, your method is correct.
  • #1
SengNee
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1) [tex]|2x+1|<4x-2[/tex]

2) [tex]|2x-1|>x+2[/tex]

3) [tex]|\frac {x-2}{x+1}|<3[/tex]

4) [tex]|2x-1|>\frac {1}{x}[/tex]


(Show me as many methods as possible. Thanks)
 
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  • #2
Each of those shows only a single absolute value in the left member. Ask yourself what happens when the expression inside the absolute value function is positive or zero; and what happens when the expression inside is less than zero. Continue with each condition to find a solution set for each exercise.
 
  • #3
Take 1:

Split up in the two cases:
A) 2x+1<4x-2 AND 2x+1>0
B) -(2x+1)<4x-2 AND 2x+1<0

Take A:
The second inequality requires x>-1/2
The first requires 3<2x, that is x>3/2

Thus, to fulfill both of these inequalities, we must have x>3/2 as the solution to A.

Now, let us tackle B:
The second inequality requires x<-1/2

The first requires:
1<6x, implying x>1/6

But these two inequalities cannot be fulfilled simultaneously, i.e, there are no solutions to case B

Thus, the entire solution to 1) is x>3/2
 
  • #4
Since you ask as many methods as possible, sketch or drwa a graph of both functions. Eg the | | part for the first two has a V-shape. It will help give you a sense of what is happening and, also as a habit, pick out mistakes sometimes.
 
  • #5
1) [tex]|2x+1|<4x-2[/tex]

[tex]2x+1<4x-2[/tex]
[tex]3<2x[/tex]
[tex]3/2<x[/tex] ... i[tex]2x+1>2-4x[/tex]
[tex]6x>1[/tex]
[tex]x>1/6[/tex] ... ii

To fulfill both i and ii, therefore [tex]3/2<x[/tex].

Can I do like that?
 
Last edited:
  • #6
You could always take the three different regions around the two critical points in case you feel confused.
 

1. How do I solve an inequality with variables on both sides?

To solve an inequality with variables on both sides, you need to isolate the variable on one side of the inequality sign. You can do this by using inverse operations, such as adding or subtracting the same number from both sides, or multiplying or dividing both sides by the same number. Once the variable is isolated, you can solve for its value.

2. What is the difference between solving an inequality and solving an equation?

The main difference between solving an inequality and solving an equation is that inequalities have a range of solutions, while equations have a single solution. Inequalities also use inequality symbols (<, >, ≤, ≥) instead of an equal sign (=).

3. Can I graph an inequality?

Yes, you can graph an inequality by first solving it and then plotting the solutions on a number line. The solutions will be shaded to indicate the range of values that satisfy the inequality. If the inequality is strict (>, <), the endpoint will be an open circle, while if it is inclusive (≥, ≤), the endpoint will be a closed circle.

4. How do I know if a solution to an inequality is inclusive or exclusive?

The type of inequality symbol used in the problem will determine if the solution is inclusive or exclusive. If the symbol is ≤ or ≥, the solution will be inclusive, meaning the endpoint is included in the solution. If the symbol is < or >, the solution will be exclusive, meaning the endpoint is not included in the solution.

5. Can I check my solution to an inequality?

Yes, you can check your solution to an inequality by plugging it back into the original inequality. If the solution makes the inequality true, then it is a valid solution. If it makes the inequality false, then it is not a valid solution. You can also graph the solution and see if it falls within the shaded region on the number line.

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