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variable in denominator

Anonymous Mon 23 Mar 11:38:18 2020 No.11494671 View Reply Original Report

Quoted By: >>11494740 >>11494749 >>11495011 >>11495027

I've been doing doing a basic bullshit math course because I have to, and I've blazed through everything but this.

This is really easy to solve graphically, but we're not allowed to. Help.

Anonymous

Anonymous Mon 23 Mar 2020 11:40:14 No.11494674 Report

Quoted By:

the fraction is positive if and only if the numerator and denominator have the same sign

Anonymous

Anonymous Mon 23 Mar 2020 11:59:33 No.11494712 Report

Quoted By:

4x+1 ---- -0.25 +++++++++++++++++++++++
3-2x ++++++++++++++++ +1.5 --------------

Anonymous

Anonymous Mon 23 Mar 2020 12:11:18 No.11494740 Report

Quoted By:

>>11494671
multiply both sides by the inverse of the denominator

Anonymous

Anonymous Mon 23 Mar 2020 12:15:02 No.11494749 Report

Quoted By: >>11494792 >>11494805

>>11494671
Multiply the inequality with (3 - 2x)^2.

(4x +1)(3 - 2x)
= -8x^2 + 10 x + 3
= -8(x^2 - 5/4 x + 25/64) + 25/8 + 3
= -8(x - 5/8)^2 + 49/8

-8(x - 5/8)^2 + 49/8 >= 0
49/64 >= (x - 5/8)^2
7/8 >= |x - 5/8|
-1/4 <= x <= 3/2

Anonymous

Anonymous Mon 23 Mar 2020 12:30:23 No.11494785 Report

Quoted By:

Google Khan Academy and look at algebra-home>rational-inequalities>one-side-is-zero

Anonymous

Anonymous Mon 23 Mar 2020 12:33:45 No.11494792 Report

Quoted By:

>>11494749
tsk tsk, remember the denominator
-1/4 <= x < 3/2

Anonymous

Anonymous Mon 23 Mar 2020 12:40:57 No.11494805 Report

Quoted By:

>>11494749
if x was equal to 3/2, the denominator would be equal to 0

Anonymous

Anonymous Mon 23 Mar 2020 14:43:22 No.11495011 Report

Quoted By:

>>11494671
The fraction nonnegative iff the numerator is zero or if the denominator and the numerator have the same sign. We assume here that the denominator is always nonzero.
Note that for $x = -1 / 4$ , we have $\frac{4x + 1}{3 - 2x} = \frac{0}{3.5} = 0$ . For any $x < 1 / 4$ , the numerator is negative and the denominator is positive since we add two positive numbers. Similarly, for $x > 3 / 2$ , the denominator is negative and the numerator is positive. We notice that equality at $3 / 2$ leads to a zero denominator, so we exclude that from our consideration. Thus, we see $-1 / 4 < x < 3 / 2$ , since the numerator and denominator are both positive in $(-1/4, 3/2)$ .

Anonymous

Anonymous Mon 23 Mar 2020 14:50:44 No.11495027 Report

Quoted By:

>>11494671 (OP)
The fraction nonnegative iff the numerator is zero or if the denominator and the numerator have the same sign. We assume here that the denominator is always nonzero.
Note that for $x = -1 / 4$ , we have $\frac{4x + 1}{3 - 2x} = \frac{0}{3.5} = 0$ . For any $x < -1 / 4$ , the numerator is negative and denominator is positive as the sum of two positive reals. Similarly for $x > 3 / 2$ , the denominator is negative and numerator positive. We notice that if $x = 3 / 2$ , we have a zero denominator, so we exclude that from our answer. We observe for $-1 / 4 < x < 3 / 2$ , both numerator and denominator are positive by use of simple inequalities, and so we conclude $x \in \left[ -1/4. 3/2 \right)$ .

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