Non-linear system of inequalities

[fluid anchor]

Correct way to solve it?
5 should be minimum since it’s the lowest point.However how to solve it without an image of the functions?

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[fluid anchor]

We could find the Lower point by comparing the y,s of the functions.Or just by the fact that that’s the vertex

[clear wind]

,w solve y >= 2(x-3)^2+5, y<=x+3

[north briar]

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Requested by tkl1

[clear wind]

,tikz[declare function={f(\x)=\x+3; g(\x)=2*(\x)^2-12*\x+23;}]
\begin{axis}[axis lines=center,grid=both,major grid style={black!50,line width=.5},minor tick num=9,minor grid style={black!40,line width=.2},xlabel={$x$},ylabel={$y$},title={Use \TeX{}it to sketch graph}]
\addplot[yellow,domain=-.5:4.5] {f(x)} node[pos=.3, above]{$y = x+3$};
\addplot[blue!60,domain=-.5:4.5] {g(x)} node[pos=.5, right]{$y = 2x^2-12x+23$};
\end{axis}

[north briar]

vin100

[clear wind]

the start of the question says "a solution". it can be a (set of) values satisfying the given system

it doesn't say "the solution", say the given (open interval) "(a, b)" doesn't not necessarily include all solutions

[steel copper]

They intersect at (4,7) and (5/2,11/2)

I don’t understand what the question is asking though

Oh is it the area in between the two things?

Cause that’s integration

[clear wind]

nah it's an inequality question

[steel copper]

Oh I see

Yeah, I hate proof, so I’m out

[clear wind]

to choose a least possible candidate b for an upper bound for the open interval (a, b) that solves the given system

[clear wind]

that's not really a proof like the ones you see in pure math. it's just a word game, playing with "a solution" and "the set of all solutions"

[steel copper]

Yh ik.

I mean the question certainly isn’t hard

It’s just I can’t remember how to do it

The top equation is in completed square form

[clear wind]

,,2(x-3)^2+5 \le y \le x+3

[north briar]

vin100

[clear wind]

forget the y in the middle, and do polynomial manipulations just like when we see the '=' sign

,calc simplify('2*(x-3)^2+5-(x+3)')

[north briar]

Result:

2 * ((x - 3) ^ 2 + 1) - x

[steel copper]

Oh yeah, I understand

[fluid anchor]

Yes

But let’s say it was some other more “complicated” function.And we don’t know what they look like

[fluid anchor]

This is what I used

But why should we assume them

What if the points of inequality didn’t lie in between the two common points

[clear wind]

it's given.
when doing question, we don't wanna miss any single given condition

[fluid anchor]

I mean in a way to sketch without actually knowing how the graph looks like

[fluid anchor]

Yes

Like let’s say we were given a third degree polynomial

To apply a similar idea to the system of inequalities

[clear wind]

i don't really get wdym

it's given

,,\begin{cases}
y &\le g(x) \ y &\ge f(x)
\end{cases}

[north briar]

vin100

vin100

[clear wind]

so $f(x) \le y \le g(x)$

[north briar]

vin100

[fluid anchor]

And we can graph them.And the that (-h,k) is the min that satisfies the inequality

What if we had other functions that we can’t graph without a software

[clear wind]

" (-h,k) is the min that satisfies the inequality" ← wdym?
(-h,k) is an open interval? are the coordinates of a point?

[fluid anchor]

(-h,k)
Satisfied every condition
And k is the min value of y that can satisfy them

But what if we didn’t know that

If it was a non parabola

How would we solve it

[clear wind]

sorry i'm a bit hungry dun wanna interpret word by word your question
in general we solve (g - f)(x) ≥ 0

[fluid anchor]

I’ll spend some more time looking at it I guess😅.Thanks