When are there 2 answers or 1 in inequalities ?

[Zulgok]

When are there 2 answers or 1 in inequalities ? For example, inequalities 3(x - 2)(x - 2.33) < 0 answers are (2; 2.33) and inequalities (x + 1)(x - 23) => 0 answers are (-~; -1][23; +~). How to know when to write one answer and when to write two ? Thank you.

 

[pka]

For example, inequalities 3(x-2)(x-2.33)<0 answers are (2;2.33) and inequalities (x+1)(x-23)=>0 answers are (-~;-1][23;+~). How to know when to write one answer and when to write two ? Thank you.

Here is a very rough idea. Recall that \(\displaystyle |a-b|\) is the distance from \(\displaystyle a~ to~ b\).
So \(\displaystyle |x+5|<3\) is the set of numbers that are less that \(\displaystyle 3\) from \(\displaystyle -5\). or \(\displaystyle (-5-3,-5+3)\).

Thus \(\displaystyle |x+5|>3\) would be outside that finite open interval: \(\displaystyle (-\infty,-8)\cup(-2,\infty)\)

 

[Ishuda]

For example, inequalities 3(x-2)(x-2.33)<0 answers are (2;2.33) and inequalities (x+1)(x-23)=>0 answers are (-~;-1][23;+~). How to know when to write one answer and when to write two ? Thank you.

In the \(\displaystyle \le\) situations
(x-a) (x-b) \(\displaystyle \le\) 0; a < b
you still must consider the two situations but one will lead to a contradiction:
(1) x-a \(\displaystyle \le\) 0 AND x-b \(\displaystyle \ge\) 0
which implies x \(\displaystyle \le\) a and x \(\displaystyle \ge\) b which is a contradiction. Since a<b, x can not at the same time be less than or equal to a and greater than or equal to b.
(2) x-a \(\displaystyle \ge\) 0 AND x-b \(\displaystyle \le\) 0
which implies x \(\displaystyle \ge\) a and x \(\displaystyle \le\) b or x \(\displaystyle \epsilon\) [a, b]. Whether the interval is open or closed depends on whether it is \(\displaystyle \le\) or just <, as you have indicated.

 

[Zulgok]

Thank you guys. I want to make sure now if i got it.

1. (x+1)(x-4)<0 is (-1;4)
2. (x+2)(x-0.5)<0 is (-2;0.5)
3. (x-1)(x+5)>0 is (-~;-5)(1;+~)
4. (x-1.5)(x+4)>0 is (-~;-4)(1.5;+~)

 

[Ishuda]

Thank you guys. I want to make sure now if i got it.

1. (x+1)(x-4)<0 is (-1;4)
2. (x+2)(x-0.5)<0 is (-2;0.5)
3. (x-1)(x+5)>0 is (-~;-5)(1;+~)
4. (x-1.5)(x+4)>0 is (-~;-4)(1.5;+~)

yep

 

[stapel]

When are there 2 answers or 1 in inequalities?

I think you're asking when, when solving quadratic inequalities, the solution will be two intervals or one interval. (Your instructor was supposed to have covered this terminology in class. If some of what I've said so far doesn't make sense or sound familiar, please do reply with questions.)

For example, inequalities 3(x - 2)(x - 2.33) < 0 answers are (2; 2.33)...

The associated function is y = 3(x - 2)(x - 7/3). Because this is a quadratic with a positive leading coefficient, when you graph this (here), the parabola will open upward. The zeroes of the function are the x-intercepts, and these zeroes split the number line (that is, the x-axis) into three sections, or intervals: (-infinity, 2), (2, 7/3), (7/3, +infinity). You can see from the graph that this positive quadratic's parabola is above the x-axis (and thus has positive y-values) for the two intervals on the ends, and is below the x-axis (and thus has negative y-values) for the interval in the middle.

Since the original inequality asks you for the interval of x-values for which this quadratic is negative, which interval(s) will be correct?

...and inequalities (x + 1)(x - 23) => 0 answers are (-~; -1][23; +~).

for this inequality, they're wanting the interval(s) for which the y-values are positive, so the associated graph is above the x-axis. Which interval(s) will be correct? (Also, this is an "or equal to" inequality, so the x-intercepts are included, too.)

 

[lookagain]

Thank you guys. I want to make sure now if i got it.

1. (x+1)(x-4)<0 is (-1;4)
2. (x+2)(x-0.5)<0 is (-2;0.5)
3. (x-1)(x+5)>0 is (-~;-5)(1;+~)
4. (x-1.5)(x+4)>0 is (-~;-4)(1.5;+~)

.

No, you didn't write the answers correctly.

Write commas instead of semicolons.

Type an appropriate symbol between the last sets of intervals in the last two
problems, such as "\(\displaystyle \cup \)." A comma might be acceptable.

[t e x]\cup[/t e x] <----- Leave out the spaces in "tex" and "/tex" for Latex.


Don't type a tilde (~) for infinity, Even "oo" approximates it well, or maybe
just type "inf" and "- inf," respectively, if you don't use Latex.

Latex appearance: \(\displaystyle \ \infty \)


[t e x]\infty[/t e x] <----- Leave out the spaces in "tex" and "/tex" for Latex.


- - - - - - - - - -- - ---- - -- - - - - - ---- - - - - - - - - - -- - - - - -- - - - - - - - - - - - - -


Example of 4 shown two ways:


(x - 1.5)(x + 4) > 0 . . . . The interval notation is (-oo, -4), (1.5, oo).

(x - 1.5)(x + 4) > 0 . . . . The interval notation is \(\displaystyle \ (- \infty, -4) \cup (1.5, \infty).\)