Question

On a coordinate plane, a solid straight line has a positive slope and goes through (negative 4, 1) and (0, 3). Everything below and to the right of the line is shaded.

Which linear inequality is represented by the graph?

y ≤ 2x + 4
y ≤ 1/2x + 3
y ≥ 1/2x + 3
y ≥ 2x + 3

Answer 1

y ≥ 2x + 3

Explanation:

Answer 2

y ≤
`(1)/(2) x+3`

Explanation:

Since everything below and to the right of the line is shaded, we must have the following:

y ≤

because the symbol ≤ implies that the area below the line is the shaded part

with this, options 3 and 4 are discarded

now, on the right side of the symbol we should have the expression for a line, which as a general form:

`mx+b`

where m is the slope and b is the y-intercept of the line

so until now the answer should be in the form

y ≤
`mx+b`

and we calculate the slope m with the two points we are given:

(-4, 1) and (0, 3)

where:

`x_(1)=-4\\y_(1)=1\\x_(2)=0\\y_(2)=3`

and we plug this values in the slope equation:

`m=(y_(2)-y_(1))/(x_(2)-x_(1)) \\\\m=(3-1)/(0-(-4))\\ \\m=(2)/(4)\\ \\m=(1)/(2)`

so now we know that our solution must have the form:

y ≤
`(1)/(2) x+b`

and b can be found since we know that the line passes through (0, 3), so when x=0 y=3, this means that the y-intercept b is 3:

y ≤
`(1)/(2) x+3`

which is the second option