Certainly! To solve the inequality [tex]\(-2x - 3y < 6\)[/tex], we can follow these steps:
1. Understand the Inequality:
The given inequality is [tex]\(-2x - 3y < 6\)[/tex]. We want to determine the set of points [tex]\((x, y)\)[/tex] in the coordinate plane that satisfy this inequality.
2. Isolate [tex]\(y\)[/tex] if Possible:
Let's try to rewrite the inequality in a form that might be more familiar. We'll solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[
-2x - 3y < 6
\][/tex]
Start by adding [tex]\(2x\)[/tex] to both sides:
[tex]\[
-3y < 2x + 6
\][/tex]
Next, divide both sides by [tex]\(-3\)[/tex]. Remember, dividing by a negative number reverses the inequality sign:
[tex]\[
y > -\frac{2}{3}x - 2
\][/tex]
3. Graphing the Inequality:
To graph this inequality, first plot the line [tex]\(y = -\frac{2}{3}x - 2\)[/tex]:
- The y-intercept is [tex]\(-2\)[/tex].
- The slope is [tex]\(-\frac{2}{3}\)[/tex], which means for every 3 units you move to the right, you move 2 units down.
Draw this line as a dashed line (because the inequality is strictly greater than, not greater than or equal to).
4. Shading the Solution Region:
Since the inequality we have is [tex]\(y > -\frac{2}{3}x - 2\)[/tex], we shade the region above the line [tex]\(y = -\frac{2}{3}x - 2\)[/tex].
5. Checking a Test Point:
To confirm that we have shaded the correct region, choose a test point not on the line, such as [tex]\((0,0)\)[/tex]:
- Plugging into the inequality [tex]\(y > -\frac{2}{3}x - 2\)[/tex] gives [tex]\(0 > -2\)[/tex], which is true.
Therefore, our shading is correct. The region that satisfies the inequality is the area above and to the left of the dashed line.
This process allows us to visualize and understand the solution set for the inequality [tex]\(-2x - 3y < 6\)[/tex] without relying on predefined scripts or specific numeric solutions.
