To determine the attributes of the boundary line of the given inequality [tex]\(-3x - 2y < 6\)[/tex], we'll transform the inequality into a more familiar form and analyze it step-by-step.
1. Rearrange the Inequality:
We start by rearranging the inequality to isolate [tex]\(y\)[/tex].
[tex]\[
-3x - 2y < 6
\][/tex]
2. Divide Both Sides by -1:
To make the inequality simpler, we can divide everything by -1. Note that dividing an inequality by a negative number flips its direction.
[tex]\[
3x + 2y > -6
\][/tex]
3. Convert to Slope-Intercept Form:
We want to write the inequality in a slope-intercept form (i.e., [tex]\(y = mx + b\)[/tex]) for easy identification of the slope and y-intercept.
[tex]\[
2y > -3x - 6
\][/tex]
To isolate [tex]\(y\)[/tex], divide everything by 2:
[tex]\[
y > -\frac{3}{2}x - 3
\][/tex]
Now, the inequality [tex]\(y > -\frac{3}{2}x - 3\)[/tex] is in slope-intercept form, where the slope [tex]\(m\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex] and the y-intercept [tex]\(b\)[/tex] is [tex]\(-3\)[/tex].
4. Type of Line:
The given inequality [tex]\( -3x - 2y < 6 \)[/tex] uses a strict inequality (less than "<"). This means the boundary line itself is not included in the solution set; hence, the boundary line will be dashed.
Here are the attributes we have determined:
- Slope [tex]\(m\)[/tex]: [tex]\(-\frac{3}{2}\)[/tex]
- Y-intercept [tex]\(b\)[/tex]: [tex]\(-3\)[/tex] (which corresponds to the point [tex]\((0, -3)\)[/tex])
- Type of Line: Dashed, because the inequality is strict ("<")
5. Correct Answer Choice:
Based on the determined attributes, the correct answer is:
[tex]\[
\text{D. The line is dashed with a } y \text{-intercept at } (0, -3) \text{ and slope of } -\frac{3}{2}.
\][/tex]
Thus, the correct answer is choice D.
