What are the solutions to the inequality?

In summary, inequality refers to the unequal distribution of resources, opportunities, and privileges among individuals or groups in a society. It can be measured using various statistical methods, including the Gini coefficient and the Human Development Index. The causes of inequality are complex and can include historical and systemic discrimination, unequal access to education and healthcare, and policies that favor certain groups. Inequality can have a significant impact on society, leading to social and economic disparities, reduced social mobility, and increased social tension, as well as contributing to issues such as poverty, crime, and health problems. To address inequality, a multifaceted approach is needed, including implementing policies that promote equal opportunities and addressing systemic discrimination, as well as increasing access to education and healthcare. It
  • #1
Darth Frodo
212
1

Homework Statement


[itex]\frac{3x + 1}{2x - 6}[/itex] < 3


Homework Equations





The Attempt at a Solution



[itex]\frac{3x + 1}{2(x -3}[/itex] < 3

[itex]\frac{3x +1}{x - 3}[/itex] < 6

Assume x < 3

3x + 1 > 6(x - 3)
3x + 1 > 6x - 18
3x + 1 - 6x + 18 > 0
19 > 3x
x < 19/3

No Contradiction.


Assume x > 3
3x + 1 < 6x - 18
3x - 19 > 0
3x > 19
x > 19/3

No Contradiction




How do I know which set of values to take?
 
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  • #2


Darth Frodo said:

Homework Statement


[itex]\frac{3x + 1}{2x - 6}[/itex] < 3

Homework Equations


The Attempt at a Solution



[itex]\frac{3x + 1}{2(x -3}[/itex] < 3

[itex]\frac{3x +1}{x - 3}[/itex] < 6

Assume x < 3

3x + 1 > 6(x - 3)
3x + 1 > 6x - 18
3x + 1 - 6x + 18 > 0
19 > 3x
x < 19/3

No Contradiction. Assume x > 3
3x + 1 < 6x - 18
3x - 19 > 0
3x > 19
x > 19/3

No Contradiction

How do I know which set of values to take?
[

Hi Darth. You need to consider both the assumption and the consequent result in determining the solution region.

For example {x < 3 and x < 19/3} means that x must be less than 19/3, but also less than 3, but this is equivalent to simply stating x < 3.
 
  • #3


For your second solution region, x must be greater than 3, but also greater than 19/3. This is equivalent to simply stating [itex] x > \ldots[/itex]. (You fill that one in. :smile:)
 
  • #4


I attempted a different method

Is the correct answer

x < 3
x > 19/3
 
  • #5


Yes that is correct.

BTW. What was your "different method". One alternative is to multiply both sides by the square of the denominator, which makes it into a single quadratic inequality.
 
  • #6


Yeah, that was what I did
Then tested the point
 
  • #7


Could also do [tex] \frac{3x+1 - 3(2x-6)}{2x-6}<0 [/tex] simplify and then draw up an algebraic table to see when each of the quantities (3x-19), (2x-6) are negative, zero or positive (or undefined)for values of x less than 3, 3, 3<x<19/3, 19/3 and greater than 19/3.
 

Related to What are the solutions to the inequality?

1. What is the definition of inequality?

Inequality is a concept that describes the unequal distribution of resources, opportunities, and privileges among individuals or groups in a society.

2. How can inequality be measured?

Inequality can be measured using various statistical methods, such as the Gini coefficient, which measures income distribution, or the Human Development Index, which looks at factors like education and life expectancy.

3. What are the causes of inequality?

The causes of inequality are complex and can vary depending on the context. Some common factors include historical and systemic discrimination, unequal access to education and healthcare, and policies that favor certain groups over others.

4. How does inequality affect society?

Inequality can have a significant impact on society, leading to social and economic disparities, reduced social mobility, and increased social tension. It can also contribute to issues such as poverty, crime, and health problems.

5. What can be done to address inequality?

Addressing inequality requires a multifaceted approach, including implementing policies that promote equal opportunities, addressing systemic discrimination, and increasing access to education and healthcare. It also requires individuals to challenge their own biases and actively work towards creating a more equitable society.

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