Sure, let's solve the inequality [tex]\( y + 2 > -3x - 3 \)[/tex] step by step, and then determine what the graph should look like.
1. Rewrite the Inequality:
To put the equation in a more familiar form, we need to isolate [tex]\( y \)[/tex].
[tex]\[
y + 2 > -3x - 3
\][/tex]
Subtract 2 from both sides:
[tex]\[
y > -3x - 3 - 2
\][/tex]
Simplify the right-hand side:
[tex]\[
y > -3x - 5
\][/tex]
2. Identify the Line:
The inequality [tex]\( y > -3x - 5 \)[/tex] involves a linear expression on the right-hand side. To graph this inequality, first, consider the related equation where the inequality is replaced by an equality:
[tex]\[
y = -3x - 5
\][/tex]
3. Graph the Line [tex]\( y = -3x - 5 \)[/tex]:
- This line has a slope of [tex]\(-3\)[/tex] and a y-intercept of [tex]\(-5\)[/tex].
- To graph it, start at [tex]\((0, -5)\)[/tex] on the y-axis.
- From the y-intercept, use the slope to find another point. A slope of [tex]\(-3\)[/tex] means that for every 1 unit you move to the right (positive x-direction), you move 3 units down (negative y-direction). So starting from [tex]\((0, -5)\)[/tex], moving to the point [tex]\((1, -8)\)[/tex] (since [tex]\(-5 - 3 = -8\)[/tex]) gives another point on the line.
4. Draw the Line:
Since the inequality is strict ([tex]\(>\)[/tex]), we use a dashed line to represent the boundary where [tex]\( y = -3x - 5 \)[/tex]. A dashed line indicates that points on the line are not included in the solution set.
5. Shade the Appropriate Region:
The inequality is [tex]\( y > -3x - 5 \)[/tex]. This means that we shade the region above the line. Any point in this region will have a y-coordinate greater than [tex]\(-3x - 5\)[/tex].
6. Conclusion:
The graph should show:
- A dashed line representing [tex]\( y = -3x - 5 \)[/tex].
- The area above this line should be shaded.
So, the characteristics of the graph we derived are:
- The slope of the line is [tex]\(-3\)[/tex].
- The y-intercept of the line is [tex]\(-5\)[/tex].
- The inequality is strict ([tex]\(>\)[/tex]), meaning the line is dashed.
- The shaded region is above the line.
These characteristics match the results:
[tex]\[
(-3, -5, '>', 'dashed')
\][/tex]
This detailed explanation should help you correctly draw and understand the graph of the inequality [tex]\( y + 2 > -3x - 3 \)[/tex].
