To find the solution to the system of inequalities
[tex]\[
\begin{cases}
x + y \le -3 \\
y < \frac{x}{2}
\end{cases}
\][/tex]
we need to graph both inequalities and identify the region where both conditions are satisfied.
### Step 1: Graph the inequality [tex]\(x + y \le -3\)[/tex]
1. Rewrite the inequality in slope-intercept form [tex]\(y \le -x - 3\)[/tex].
2. The equality [tex]\(y = -x - 3\)[/tex] is a straight line with a slope of -1 and a y-intercept of -3.
To graph this:
- The y-intercept is the point [tex]\((0, -3)\)[/tex].
- The slope of -1 means that for every unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by 1.
Plot another point by changing [tex]\(x\)[/tex]. For example:
- If [tex]\(x = 1\)[/tex], then [tex]\((1, -4)\)[/tex].
Draw the line passing through these points. As the inequality is [tex]\(\le\)[/tex], we fill in the region below and including the line.
### Step 2: Graph the inequality [tex]\(y < \frac{x}{2}\)[/tex]
1. Rewrite the inequality in slope-intercept form (it's already there) [tex]\(y < \frac{x}{2}\)[/tex].
2. The equality [tex]\(y = \frac{x}{2}\)[/tex] is a straight line with a slope of [tex]\( \frac{1}{2} \)[/tex] and a y-intercept of 0.
To graph this:
- The y-intercept is the point [tex]\((0, 0)\)[/tex].
- The slope [tex]\(1/2\)[/tex] means that for every unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by 0.5.
Plot another point by changing [tex]\(x\)[/tex]. For example:
- If [tex]\(x = 2\)[/tex], then [tex]\((2, 1)\)[/tex].
Draw the line passing through these points. Since the inequality is [tex]\(<\)[/tex], we fill in the region below the line with a dashed line to show that the line itself is not included.
### Step 3: Identify the solution region
The solution region for the system of inequalities is the overlap of the two shaded regions from Steps 1 and 2:
- The region below the line [tex]\(y = -x - 3\)[/tex] (including the line).
- The region below the line [tex]\(y = \frac{x}{2}\)[/tex] (not including the line itself).
### Step 4: Sketch the combined graph
- Draw the line [tex]\(y = -x - 3\)[/tex], shading the region below it.
- Draw the line [tex]\(y = \frac{x}{2}\)[/tex] (with a dashed line), shading the region below it.
- The intersection (overlap) of these two shaded regions is the solution to the system.
Here's what the final graph should look like:
1. A solid line for [tex]\(y = -x - 3\)[/tex] with the region below it shaded.
2. A dashed line for [tex]\(y = \frac{x}{2}\)[/tex] with the region below it shaded.
3. The solution region is where the two shaded areas overlap. It's effectively the area below [tex]\(y = -x - 3\)[/tex] and simultaneously below or to the left of [tex]\(y = \frac{x}{2}\)[/tex].
By following these steps, you can accurately graph and find the solution region for the given system of inequalities.
