Math Snap

STEP 1

What is this asking?
We need to draw the line for $3x - 2y = 18$ and then shade the region that represents $3x - 2y < 18$.
Watch out!
Remember that when we multiply or divide both sides of an inequality by a negative number, we must flip the inequality sign!
Also, a dashed line means points on the line are not included in the solution.

STEP 2

1. Rewrite the inequality as a linear equation.
2. Find two points that satisfy the equation.
3. Graph the line using the two points, remembering to use a dashed line.
4. Choose a test point not on the line.
5. Check if the test point satisfies the original inequality.
6. Shade the correct region.

STEP 3

Let's rewrite our inequality $3x - 2y < 18$ as an equation to find the boundary line: $3x - 2y = 18$.
This helps us visualize the line that separates the solution region from the non-solution region.

STEP 4

To graph the line, we need two points.
Let's pick $x = 0$.
Substituting this into our equation $3x - 2y = 18$, we get $3 \cdot 0 - 2y = 18$, which simplifies to $-2y = 18$.
Dividing both sides by $-2$ gives us $y = -9$.
So, our first point is $(0, -9)$.

STEP 5

Now, let's pick $y = 0$.
Substituting into $3x - 2y = 18$, we get $3x - 2 \cdot 0 = 18$, which simplifies to $3x = 18$.
Dividing both sides by $3$ gives us $x = 6$.
So, our second point is $(6, 0)$.

STEP 6

Plot the two points $(0, -9)$ and $(6, 0)$ on the coordinate plane.
Since our original inequality is $3x - 2y < 18$ (strictly less than), we draw a dashed line through these points.
The dashed line indicates that points on the line are not part of the solution.

STEP 7

A convenient test point is the origin $(0, 0)$ because it makes calculations easy!
Make sure your test point isn't on the line we just drew.

STEP 8

Substitute the test point $(0, 0)$ into the original inequality $3x - 2y < 18$.
We get $3 \cdot 0 - 2 \cdot 0 < 18$, which simplifies to $0 < 18$.
This is true!

STEP 9

Since our test point $(0, 0)$ satisfies the inequality, we shade the region that contains the origin.
This is the region above the dashed line.
If the test point had not satisfied the inequality, we would have shaded the opposite side of the line.

The solution is the region above the dashed line $3x - 2y = 18$, including all points within that region but not the points on the line itself.