Math Snap

STEP 1

What is this asking?
We need to find the equation of the line, and then figure out which side of the line is shaded to write the inequality.
Watch out!
Make sure to use the correct inequality symbol!
Since the line is solid, we'll use either $\le$ or $\ge$.

STEP 2

1. Find the equation of the line.
2. Determine the inequality.

STEP 3

Alright, let's find the slope!
We've got two points: $(-8, 8)$ and $(8, -8)$.
The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
Plugging in our points, we get $m = \frac{-8 - 8}{8 - (-8)} = \frac{-16}{16} = -1$.
So, our slope is $-1$!

STEP 4

Now for the y-intercept!
We can use the point-slope form: $y - y_1 = m(x - x_1)$.
Let's use the point $(-8, 8)$ and our slope $m = -1$.
We get $y - 8 = -1(x - (-8))$, which simplifies to $y - 8 = -x - 8$.
Adding $8$ to both sides, we find $y = -x + 0$, so $y = -x$.
Our y-intercept is $0$!

STEP 5

So, the equation of the line is $y = -x$.

STEP 6

The shaded region is below the line.
Let's pick a test point in the shaded region, like $(0, -1)$.

STEP 7

If we plug in $(0, -1)$ into our equation, we get $-1 = -0$, which simplifies to $-1 = 0$.
That's not true!
We need an inequality, so we ask ourselves: is $-1$ less than or greater than $0$?
It's less than!
So, we use the $\le$ symbol.

STEP 8

Since the line is solid, we use $\le$, not $<$.
Therefore, our inequality is $y \le -x$.

The inequality is $y \le -x$.