Question

Which of the following points lie in the solution set to the following system of inequalities?

y less than or greater to x - 5
y less than or greater to -x - 4

(1, 10)

(-1, 10)

(10, 1)

(1, -10)

Answer 1

just like the ones before, test each choice, see which one gives true for both inequalities

Answer 2

What do you mean? @jdoe0001

Answer 3

we have done these ones before
about 2 or 3 days ago IIRC

Answer 4

say for example testing the 1st point
(1, 10)
x y

(10) <= (1)-5 => 10 <= -4

so is 10 really less than -4? well that's not true, thus is not a solution

and then test the other points

Answer 5

I don't understand what you just did. @jdoe0001

Answer 6

hmm ----> y less than or greater to x - 5
is this supposed to be -> y less than or EQUAL to x - 5?

Answer 7

Equal to.

Answer 8

so in the system of equations \(\bf y \le x-5\\
y \le -x-4\)

you use the 2nd point and test, and then the 3rd and test and then the 4th one
the solution will be where the inequalities become true

as you see above, I replace my (x, y) = (1, 10), so x = 1, y =10 in the 1st equation
I ended up with 10 <= -4
but of course 10 is not less than -4, thus is not true thus is not a solution

Answer 9

So you replaced (x, y) = (1, 10) and you got 10 <= -4?

Answer 10

yes

Answer 11

of course 10 is not less than -4, thus is not true thus is not a solution

Answer 12

so the solution point, will give TRUE on both equations when replacing the (x,y) for the given point

something TRUE will be say 3 < 6, or 2 < 5, or -1 > -5 or 100 > 99
something FALSE will be say 1 > 2, or 3 < 1 and such

Answer 13

they're false because 1 is not greater than 2, and 3 is not less than 1

Answer 14

3< 6 is true because, indeed, 6 is really greater than 3, by 3 units
and 3 is indeed smaller than 6

Answer 15

Yeah I know that but how exactly do you solve this? You replace (x, y) = (1, 10) but then what?

Answer 16

(1, 10)
x y

(10) <= (1)-5 => 10 <= -4

then expand and add each side

Answer 17

How do you expand and add each side?

Answer 18

same way you'd simplify any terms

Answer 19

all I did was replace the x and y for the values given in the point
and use those values in the x and y in the equation

Answer 20

Can you show me that in equation form instead of words?

Answer 21

@jdoe0001

Answer 22

(1, 10)
x y
\(\bf y \le x-5\)
(10) <= (1)-5 => 10 <= -4

Answer 23

Could you explain how you got the second part of that last line?

Answer 24

@jdoe0001

Answer 25

=> 10 <= -4

Answer 26

\(\bf (\color{blue}{10}) <= (\color{green}{1})-5 => \color{blue}{10} <= \color{green}{1}-4\)

Answer 27

meh

Answer 28

\(\bf (\color{blue}{10}) <= (\color{green}{1})-5 => \color{blue}{10} <= \color{green}{1}-5\)

Answer 29

1-5 = -4

Answer 30

Could you explain it?

Answer 31

hmmm what's confusing you?

Answer 32

(10)<=(1)−5=>10<=1−5 What are you doing there?

Answer 33

\(\bf (\color{blue}{10}) \le (\color{green}{1})-5 \implies \color{blue}{10} \le \color{green}{1}-5\)

Answer 34

just adding, nothing else

Answer 35

Adding the whole thing?

Answer 36

nope adding each side by itself

Answer 37

And how do you do that? What exactly are you adding?

Answer 38

if the expression were for example
lemme give you a different inequality and point

y + 2y <= 3x -2x+5
point will be ( 5, 7 )
x y

(7) + 2(7) < = 3(5)-2(5)+5 # all y = 7, x = 5 thus

7 + 14 <= 15-10+5
21 <= 10

now is 21 really less than 10? well then is a false inequality
that means the point (5, 7) is not a solution for that inequality

Answer 39

# all y = 7, x = 5 thus

What?

Answer 40

y = 7, x = 5

Answer 41

is all

Answer 42

Okay, thank you.

Answer 43

yw