Question

i need help finding the inequality in a graph. my teacher didn't explain it very well. please can someone explain?

Answer 1

The first step is to find the equation of the "dotted line" they show in the graph.
Do you know how to do that ?

Answer 2

@phi so thats the x2-x1/y2-y1?

Answer 3

that (sort of) looks like how to find the slope (but it's upside down)

but to find the equation of a line you first need
1) two points on the line
then find the slope using change in y divided by change in x

they give you two points, so that part is easy
can you find the slope ?

Answer 4

1/5?

Answer 5

ok, (-1 - -2 ) / (5 - 0) or (-1+2)/5 or 1/5
yes, the slope is 1/5
next write
y = mx + b
put in 1/5 for m


y =(1/5) x + b

now use one of the points (e.g. (0,-2) to find b)
-2= (1/5)*0 + b
what is b ?

Answer 6

b is -2 then since thats where it intercepts the y axis

Answer 7

yes, or if we simplify
-2= (1/5)*0 + b
we get
-2 = b

so the equation is
y= (1/5)x - 2

next, we see the region below this line is shaded.
if we keep "x" constant, for example x=0
all the y values below the line (at x=0) are more negative (i.e. smaller) than the y value of the line. Does that make sense ?

Answer 8

For example the point (0, -3) is in the green shaded region
and -3 is < -2 (of the y value of the line)

Answer 9

a bit, basically the y value will always be smaller than x?

Answer 10

when on the line that is

Answer 11

no. the idea is pick any x number on the x-axis. example x=5
now all the points at x=5 are either above the line or below the line
i.e. as we move straight up/down along the line x=5, some points are above the line
y= (1/5)x -2 and some are below the line

Answer 12

if we want the points below the line y=(1/5)x-2
we want y < (1/5)x-2

Answer 13

maybe doing this helps:

is the point (0,0) above the line y= (1/5)x-2 or below it ?

Answer 14

above

Answer 15

and if we use the equation
y=(1/5)*0 -2 we get y=-2 for the y value of the line
but the y value of (0,0) is 0 > -2
which tells us that (0,0) is above the line

in your problem, we want the points where the y value of a point is below the y value of the line (at that x)

anyway, the answer is
y < (1/5) x -2
or
(if they write it a different way)
-x + 5y < -10
or
x - 5y > 10 (this is tricky!)

Answer 16

btw, the dotted line means we use < or >
(the points on the line are not part of the region we want)

if the line were solid, then we use \( \le\) or \(\ge\) and the points on the line are included as part of the region we want.