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Compound Linear Inequality











































A compound linear inequality is an inequality which combines two (or more) separate linear inequalities.

A compound inequality can be formed in two different ways:


1. The word "and" or the logic symbol ∧ can be used to combine separate inequalities into a compound inequality. In this case the compound inequality is only true when all of the individual inequalities which have been joined together are true.

2. The word "or" or the logic symbol ∨ can be used to combine separate inequalities into a compound inequality. In this case the compound inequality is true when any one of the individual inequalities which have been joined together is true.


Write the solution to the following compound inequality in interval notation:

4y + 2 ≥ 10 OR 3y - 3 > 12


Comments for Compound Linear Inequality

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May 06, 2013
Compound Inequality
by: Staff


Answer

Part I


The compound inequality

Compound Inequality - 4y + 2 ≥ 10 OR 3y - 3 > 12





Two inequalities must be solved: the inequality on the left of the "or" symbol, and the inequality on the right of the "or" symbol.


the inequality on the left of the "or" symbol

subtract 2 from each side of the inequality

Compound Inequality - left 
   4y + 2 ≥ 10 
  subtract 2 from each side of the inequality





divide each side of the inequality by 4

Compound Inequality - left 
   4y  ≥ 8 
  divide each side of the inequality by 4





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May 06, 2013
Compound Inequality
by: Staff


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Part II


the inequality on the right of the "or" symbol

add 3 to each side of the inequality

Compound Inequality - right 
   3y - 3 > 12 
  add 3 to each side of the inequality





divide each side of the inequality by 3

 Compound Inequality - left 
   3y  > 15 
  divide each side of the inequality by 3





With the left and right inequalities solved, the compound inequality now looks like this:

Compound Inequality - left  and right inequalities solved






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May 06, 2013
Compound Inequality
by: Staff


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Part III


Final Answer

Since (y ≥ 2 ) includes the values represented by ( y > 5), the final solution to the compound inequality is (y ≥ 2 )

Compound Inequality - final solution:  y  ≥ 2





the final solution of valid y values in INTERVAL NOTATION is:

Interval Notation - Compound Inequality - final solution:  y  ≥ 2




∈ = element of a set

Left “[” = closed, end value (the minimum value) is INCLUDED

Right “)” = open, end value (the maximum value) is NOT INCLUDED

[2, ∞)= half open interval, left value of 2 is INCLUDED, right value of ∞ is NOT INCLUDED

a graph of the solution in interval notation is shown below:

Graph of the solution set in interval notation







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May 06, 2013
Compound Inequality
by: Staff


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Part IV


the final solution of valid y values in SET BUILDER Notation is:

There is more than one format for writing this sequence in Set Builder Notation.


Here is a commonly used format:


the final solution (set Z) of valid y values in SET BUILDER NOTATION is:

Set builder notation of solution to  compound inequality




{} curly brackets surround the expression, indicating “this is a set”

∈ = element of a set

| and : can be used interchangeably. Both notations are separators which mean “where” or “such that”

y: the first “y” is the “output function”, shown as = {y |

y: the second “y” is the “variable”, shown as | y ∈ ℝ,

ℝ (the set of all real numbers) is the "input set”

y ≥ 2 is the “predicate”


Reading from left to right: “Z” is the set of all numbers “y” {y | … , …} where “y” is an element of the set of real numbers ℝ { … | y ∈ ℝ, … } and “y” is greater than or equal to 2 {… | … , y ≥ 2 }.






Thanks for writing.

Staff
www.solving-math-problems.com


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