Question

Can someone please explain to me how i can write a system of inequalities to represent a shaded part on my graph? I only have the graph to work with. Where should i start?

Answer 1

@3mar @AloneS @KamiBug @pooja195

Answer 2

do you know what area has to be shaded ?

Answer 3

the graph that i have shows the 2 lines, were they intersect and the shaded region

Answer 4

ok

In that case, you need to figure out the equation of each line.
Pick one of the lines.

Can you find two points on the line (that are easy to read off the graph ) ?

Answer 5

yes i can. the first line has the points 3,1 and 0,4

Answer 6

ok, find the slope (change in y divided by change in x)
we already know 4 is the y-intercept (from the point 0,4 )
the equation will be
y = mx + 4

where m is the slope you find.

Answer 7

okay so y = -1x + 4

Answer 8

yes, or just
y= -x+4 (we don't usually bother to write the 1)

next, is the point (0,0) part of the shaded region ?

Answer 9

umm well the second line passes through the point 0,0

Answer 10

Maybe I should have asked: is the shaded region "below" the line ?

Answer 11

yes the shaded region is below the first line

Answer 12

also, is at the line solid or dotted ?

Answer 13

the first line is solid

Answer 14

shaded below means we want all y values "beneath" the line
y < -x+4 (if the line was dashed we use a less than)
\( y \le -x+4 \) if the line is solid (we use less than or equal)

as a check, point 0,0 is beneath the line, so the relation should be true:
putting in 0 for both x and y we get
0 <= 0+4
0<= 4 which is true. So we have the first equation

we now do the second line the same way:
1) find two points
2) find the slope
3) find the y-intercept
4) decide if we want the y's above (use > or >=) or below (use < or <=) the line

Answer 15

okay so for the second line, its dotted, the shaded is above the line,
as for the points, the only ones that i can see are 3,1 . 0,0 . and -3, -1

Answer 16

you only need two points to define a line. 0,0 and 3,1 look easy
change in y is 1-0
change in x is 3-0
and the slope is ⅓
the y-intercept is 0
if we did not have a nice point like 0,0, we would find the y-intercept like this:

write:
y= ⅓ x + b
using any point, say, 3,1 put in 3 for x and 1 for y
1= ⅓ * 3 + b
1= 1+b
add -1 to both sides to get
1-1= 1-1+b or
0 = 0+b
0= b
and b = 0

the equation is
y= ⅓ x + 0
but of course we don't bother to write +0

y= ⅓ x

we want the points above the line i.e. we want all the y's bigger than the line's y
so change the = to >

Answer 17

so
y < -x + 4
and
y > 1/3x

is the system of inequalities?

Answer 18

almost. I thought the first line was solid. the "solid" means the line is part of the solution
we use \( \le \)

See attached for what this system looks like.

Answer 19

yes that looks just like the graph i have. and i know its supposed to have the line underneath but i wasnt sure how to make it on here.

Answer 20

Thank you so much for your help!!! :D

Answer 21

btw, people use <= for \( \le \)
you read it as "less than or equal"

Answer 22

ah okay that works

Answer 23

I am sorry @MysticFury for late!
The electricity was off!

With @phi, you are in good hands!
Thanks for him for good explanation.