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Quadratic Inequality Help

No idea why I'm getting this wrong.. but..

(2x+3)2≤81(2x+3)^2 \leq 81

4x2+12x−72≤04x^2 + 12x -72 \leq 0

x2+3x−18≤0x^2 + 3x -18 \leq 0

(x+6)(x−3)≤0(x+6)(x-3) \leq 0

x≤−6x \leq -6

x≤3x \leq 3

Yet when I put it into autograph I get this:



What gives? >.<
(edited 9 years ago)
dN6C5FZ.png3.3KB
Should be a combined inequality since it's less than/equal to zero
Reply 2
Avatar for SamKeene
SamKeene
OP
11
Original post by Jackasnacks
Should be a combined inequality since it's less than/equal to zero


So yeah I consulted my book and you're right, silly mistake... Is it correct that I have to interpret the type of symbol such as ≤\leq from a geometrical look at f(x)? Cannot it be done purely algebraically?
Reply 3
Avatar for james22
james22
16
Original post by SamKeene
So yeah I consulted my book and you're right, silly mistake... Is it correct that I have to interpret the type of symbol such as ≤\leq from a geometrical look at f(x)? Cannot it be done purely algebraically?


If you don't want to use the graph, consider the various cases that arise seperately. Here it would be x>=3, -6<=x<=3, x<=-6.
Original post by SamKeene
So yeah I consulted my book and you're right, silly mistake... Is it correct that I have to interpret the type of symbol such as ≤\leq from a geometrical look at f(x)? Cannot it be done purely algebraically?


Well yes but a quick 3 second sketch of the graph shows whether it's above/below axis
It should then be clear whether it's two separate equalities or a combined one
Reply 5
Avatar for davros
davros
16
Original post by SamKeene
So yeah I consulted my book and you're right, silly mistake... Is it correct that I have to interpret the type of symbol such as ≤\leq from a geometrical look at f(x)? Cannot it be done purely algebraically?


Yes - if you want a result that is negative then the brackets must have opposite signs.
Original post by SamKeene
No idea why I'm getting this wrong.. but..

(2x+3)2≤81(2x+3)^2 \leq 81

4x2+12x−72≤04x^2 + 12x -72 \leq 0


A minor point: here you can write (2x+3)2≤81⇒−9≤2x+3≤9(2x+3)^2 \le 81 \Rightarrow -9 \le 2x+3 \le 9 directly, which removes the need for factorisation.

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