Precalc project: positive values in an inequality

So, you just have to find which of those sets are correct. If you do not understand how to solve the inequality (which is not calculus, it is precalculus), you can just try each of the 5 sets and see which one works (which one satisfies the inequality).
  • #1
hpdwnsn95
6
0

Homework Statement


Determine the set of positive values of x that satisfy the following inequality:
(1/x) - (1/(x-1)) > (1/(x-2))

a) (0, 1) union (2^1/2, 2) b) (0, 1/2) union (1, 2) c) (1/2, 1) union (2^1/2, 2(2^1/2))
d) (0, 2^1/2) union (3/2, 2) e) (1, 2^1/2) union (2, 2(2^1/2))

Homework Equations


i'm not sure that there are any relevant equations, just started precalc a few weeks ago and I'm not sure how to solve the problem


The Attempt at a Solution


I've tried graphing the equation and then tracing x to find y, but I haven't been able to find an answer that way. I've also tried solving for x which i get 1/2 > x for, but I'm not sure if that applies to this problem
 
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  • #2
hpdwnsn95 said:

Homework Statement


Determine the set of positive values of x that satisfy the following inequality:
(1/x) - (1/(x-1)) > (1/(x-2))

a) (0, 1) union (2^1/2, 2) b) (0, 1/2) union (1, 2) c) (1/2, 1) union (2^1/2, 2(2^1/2))
d) (0, 2^1/2) union (3/2, 2) e) (1, 2^1/2) union (2, 2(2^1/2))

Homework Equations


i'm not sure that there are any relevant equations, just started precalc a few weeks ago and I'm not sure how to solve the problem


The Attempt at a Solution


I've tried graphing the equation and then tracing x to find y, but I haven't been able to find an answer that way. I've also tried solving for x which i get 1/2 > x for, but I'm not sure if that applies to this problem
The first step would be to combine the two fractions on the left side.
 
  • #3
hpdwnsn95 said:

Homework Statement


Determine the set of positive values of x that satisfy the following inequality:
(1/x) - (1/(x-1)) > (1/(x-2))

a) (0, 1) union (2^1/2, 2) b) (0, 1/2) union (1, 2) c) (1/2, 1) union (2^1/2, 2(2^1/2))
d) (0, 2^1/2) union (3/2, 2) e) (1, 2^1/2) union (2, 2(2^1/2))

Homework Equations


i'm not sure that there are any relevant equations, just started precalc a few weeks ago and I'm not sure how to solve the problem


The Attempt at a Solution


I've tried graphing the equation and then tracing x to find y, but I haven't been able to find an answer that way. I've also tried solving for x which i get 1/2 > x for, but I'm not sure if that applies to this problem

Is this actually a multiple choice question? Is that what the a) b) c) etc refer to ?
 
  • #5
hpdwnsn95 said:

Homework Statement


Determine the set of positive values of x that satisfy the following inequality:
(1/x) - (1/(x-1)) > (1/(x-2))

a) (0, 1) union (2^1/2, 2) b) (0, 1/2) union (1, 2) c) (1/2, 1) union (2^1/2, 2(2^1/2))
d) (0, 2^1/2) union (3/2, 2) e) (1, 2^1/2) union (2, 2(2^1/2))

Homework Equations


i'm not sure that there are any relevant equations, just started precalc a few weeks ago and I'm not sure how to solve the problem


The Attempt at a Solution


I've tried graphing the equation and then tracing x to find y, but I haven't been able to find an answer that way. I've also tried solving for x which i get 1/2 > x for, but I'm not sure if that applies to this problem

Remember that if this is a multiple choice question, you do not have to find the answer, you only have to identify which of the 5 offerings is the answer.

While not a great method of solving a problem, substituting the offerings into the original inequality may be the fastest way for you to solve this.
 
  • #6
hpdwnsn95 said:

Homework Statement


Determine the set of positive values of x that satisfy the following inequality:
(1/x) - (1/(x-1)) > (1/(x-2))

a) (0, 1) union (2^1/2, 2) b) (0, 1/2) union (1, 2) c) (1/2, 1) union (2^1/2, 2(2^1/2))
d) (0, 2^1/2) union (3/2, 2) e) (1, 2^1/2) union (2, 2(2^1/2))

Homework Equations


i'm not sure that there are any relevant equations, just started precalc a few weeks ago and I'm not sure how to solve the problem


The Attempt at a Solution


I've tried graphing the equation and then tracing x to find y, but I haven't been able to find an answer that way. I've also tried solving for x which i get 1/2 > x for, but I'm not sure if that applies to this problem

If you do not know or see a clever trick to use, then solving the problem using basic principles is the best way. In the sample problem, start by adding the additive inverse of the righthand member to both sides; then perform the arithmetic steps for the rational expressions. You have some rational expression related to zero. Now just focus on the numerator.
 
  • #7
Given that x is positive, I would separate this problem into three parts:
x between 0 and 1: both x- 1 and x- 2 are negative.

x between 1 and 2: x- 1 is positive but x- 2 is negative

x greater than 2: both x- 1 and x- 2 are positive.

For each of those, multiply the entire inequality by x(x-1)(x-2).
 
  • #8
PeterO said:
Is this actually a multiple choice question? Is that what the a) b) c) etc refer to ?

yes it is
 
  • #9
PeterO said:
Remember that if this is a multiple choice question, you do not have to find the answer, you only have to identify which of the 5 offerings is the answer.

While not a great method of solving a problem, substituting the offerings into the original inequality may be the fastest way for you to solve this.

I would do that but I don't understand the answer format, i"ve never seen union answers before, can you (or anyone else) expain them?
 
  • #10
Do you know what the definition of the union of two sets is?
 
  • #11
SammyS said:
Do you know what the definition of the union of two sets is?

i think union stands for and right? so it's like one set & one set
but I'm not sure how would plug either set into the problem because there's no y in the problem
 
  • #12
One piece (interval) of the x-axis plus (as in union) another piece. --- Actually, all the numbers in one interval together with all the numbers in another interval together with ...
 
  • #13
hpdwnsn95 said:
I would do that but I don't understand the answer format, i"ve never seen union answers before, can you (or anyone else) expain them?

The first option

(0, 1) union (2^1/2, 2)

suggests it is true for any value between 0 and 1, as well as values between √2 and 2, but false for values between 1 and √2, and false for values above 2
it is clearly undefined for x = 0,1 or2 due to a fraction becoming 1/0

You could substitute 0.5 [1/2], and check it is true, 1.25 [5/4] and check it is false, 1.5 [3/2] and check it is true and 3 to check it is false.

Solving is better, but substitution could be your last resort.
 

FAQ: Precalc project: positive values in an inequality

What are positive values in an inequality?

Positive values in an inequality refer to any numbers that are greater than zero. In an inequality equation, positive values are represented by numbers on the right side of the inequality sign (> or <), while negative values are represented on the left side.

How do you solve an inequality with positive values?

To solve an inequality with positive values, you need to isolate the variable on one side of the equation. This is done by using inverse operations, such as addition, subtraction, multiplication, and division, to cancel out any numbers or variables on the same side of the inequality sign. Once the variable is isolated, you can determine the range of positive values that will make the inequality true.

What is the importance of positive values in an inequality?

Positive values in an inequality are important because they help us understand the range of values that will make an inequality true. They also allow us to compare two quantities and determine which one is larger or smaller. In real-life situations, positive values in inequalities can help us make decisions about things like budgeting, planning, or setting goals.

Can an inequality have more than one positive value?

Yes, an inequality can have multiple positive values. In fact, most inequalities have an infinite number of positive values that satisfy the inequality. For example, in the inequality x > 5, any value of x that is greater than 5, such as 6, 10, or 100, would be considered a positive value that makes the inequality true.

How do positive values in an inequality relate to real-life scenarios?

Positive values in an inequality relate to real-life scenarios in many ways. For example, when budgeting for a trip, we may use an inequality to determine the maximum amount of money we can spend each day. The positive values in the inequality would represent the different amounts we could spend and still stay within our budget. In economics, inequalities are often used to show the relationship between supply and demand, with positive values representing the prices at which a product will be supplied or demanded.

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