Math Snap

STEP 1

What is this asking?
We need to graph two inequalities on the same coordinate plane and find the overlapping shaded region!
Watch out!
Don't forget to check if the lines themselves are included in the solution, based on whether the inequality is "less than or equal to" or just "less than".

STEP 2

1. Graph the first inequality
2. Graph the second inequality
3. Find the overlapping region

STEP 3

Alright, let's start with the first inequality: $y > -x - 4$.
Imagine it as an equation for a moment: $y = -x - 4$.
This is a line!
Its slope is $-1$ and its y-intercept is $-4$.

STEP 4

To graph this line, we can start by plotting the y-intercept, which is the point $(0, -4)$.
From there, we use the slope!
A slope of $-1$ means that for every 1 unit we move to the right along the x-axis, we move 1 unit down along the y-axis.
So, from $(0, -4)$, we can plot another point at $(1, -5)$.

STEP 5

Now, draw a dashed line through these points.
It's dashed because our inequality is $y > -x - 4$, not $y \geq -x - 4$, meaning the line itself isn't included in the solution.

STEP 6

Time to shade!
Since the inequality is $y > -x - 4$, we want to shade the area above the line.
Pick a test point not on the line, like $(0, 0)$.
Is $0 > -0 - 4$?
Yes! $0 > -4$.
So, we shade the side of the line that includes $(0, 0)$.

STEP 7

On to the second inequality: $y \leq \frac{1}{4}x + 1$.
Again, think of it as an equation: $y = \frac{1}{4}x + 1$.
Here, the slope is $\frac{1}{4}$ and the y-intercept is $1$.

STEP 8

Plot the y-intercept at $(0, 1)$.
The slope of $\frac{1}{4}$ means that for every 4 units we move right, we move 1 unit up.
So, from $(0, 1)$, we can plot another point at $(4, 2)$.

STEP 9

Now, draw a solid line through these points.
It's solid because our inequality is $y \leq \frac{1}{4}x + 1$, meaning the line itself is included in the solution.

STEP 10

For shading, since it's $y \leq \frac{1}{4}x + 1$, we shade below the line.
Using $(0, 0)$ as a test point again: Is $0 \leq \frac{1}{4}(0) + 1$?
Yes! $0 \leq 1$.
So, we shade the side of the line that includes $(0, 0)$.

STEP 11

The solution to the system of inequalities is the region where the shading from both inequalities overlaps!
This is the region that satisfies both $y > -x - 4$ and $y \leq \frac{1}{4}x + 1$ simultaneously.

The solution is the overlapping shaded region on the graph, including the solid line from $y \leq \frac{1}{4}x + 1$ within the overlapping region, but excluding the dashed line from $y > -x - 4$.