Quadratic Inequality 2

[mathdad]

Solve the inequality.

x^2 - x - 6 > 0

Replace > with the = symbol.

x^2 - x - 6 = 0

Factor.

(x - 3)(x + 2) = 0

Solve each factor for x.

x = 3 and x = - 2

Pick a number between
-2 & 3.

Let x = 0

x^2 - x - 6 > 0

(0)^2 - 0 - 6 > 0

- 6 > 0...false statement.

I understand this to mean that the given function DOES NOT have a solution on the interval
(-2, 3).

I conclude from this finding that x^2 - x - 6 is greater than 0 in the intervals
(-infinity, -2) and (3, infinity).

Note:

Do I say (-infinity, -2) and (3, infinity)?
Should I say (-infinity, - 2) or (3, infinity)?

 

[MarkFL]

I would use interval notation:

[MATH](-\infty,-2)\,\cup\,(3,\infty)[/MATH]
This is the union of the two intervals making up the solution, which is an AND rather than an OR.

 

[Dr.Peterson]

Do I say (-infinity, -2) and (3, infinity)?
Should I say (-infinity, - 2) or (3, infinity)?

Either "and" or "or" is easily misunderstood here.

Properly, you need to use "union", because the solution is the union of the two sets.

We often use "or" here, because we are talking about any value of x that is in this set OR that set. But students can too easily take that to mean that "either this set or that set" is the solution, and in a sense it is both sets that together are the solution, making it sort of an AND.

When you use interval notation, which is a representation of sets, use the union symbol, which is a way to combine sets.

When you just use inequalities, use OR: x < -2 OR x > 3.

 

[mathdad]

Either "and" or "or" is easily misunderstood here.

Properly, you need to use "union", because the solution is the union of the two sets.

We often use "or" here, because we are talking about any value of x that is in this set OR that set. But students can too easily take that to mean that "either this set or that set" is the solution, and in a sense it is both sets that together are the solution, making it sort of an AND.

When you use interval notation, which is a representation of sets, use the union symbol, which is a way to combine sets.

When you just use inequalities, use OR: x < -2 OR x > 3.

Thank you for the information.

 

[Steven G]

I would use interval notation:

[MATH](-\infty,-2)\,\cup\,(3,\infty)[/MATH]
This is the union of the two intervals making up the solution, which is an AND rather than an OR.

Mark, I am sorry but actually no x value is in [MATH](-\infty,-2)\,and(3,\infty)[/MATH]. It should be [MATH](-\infty,-2) OR (3,\infty)[/MATH]

 

[MarkFL]

Mark, I am sorry but actually no x value is in [MATH](-\infty,-2)\,and(3,\infty)[/MATH]. It should be [MATH](-\infty,-2) OR (3,\infty)[/MATH]

What I meant by that, is that the interval \((-\infty,-2)\) AND the interval \((3,\infty)\) are part of the solution set. In retrospect, I should have avoided that part of it altogether, as there is too much room for confusion.