Graph the solution set of the following system of inequalities. $$ \begin{aligned} x+2 y & \leq 4 \ y & \geq x^{2}-3\end{aligned} $$

Question

UpStudy AI · Accepted Answer

To graph the solution set of the system of inequalities given by:

1. $$ x + 2y \leq 4 $$
2. $$ y \geq x^2 - 3 $$

we will follow these steps:

### Step 1: Graph the first inequality $$ x + 2y \leq 4 $$

1. **Convert to slope-intercept form**:
   $$
   2y \leq -x + 4 \implies y \leq -\frac{1}{2}x + 2
   $$

2. **Graph the line** $$ y = -\frac{1}{2}x + 2 $$:
   - The y-intercept is $$ (0, 2) $$.
   - The x-intercept can be found by setting $$ y = 0 $$:
     $$
     0 = -\frac{1}{2}x + 2 \implies \frac{1}{2}x = 2 \implies x = 4 \quad \text{(x-intercept is (4, 0))}
     $$

3. **Draw the line** and shade below it (since $$ y \leq -\frac{1}{2}x + 2 $$).

### Step 2: Graph the second inequality $$ y \geq x^2 - 3 $$

1. **Graph the parabola** $$ y = x^2 - 3 $$:
   - The vertex of the parabola is at $$ (0, -3) $$.
   - The parabola opens upwards.

2. **Draw the parabola** and shade above it (since $$ y \geq x^2 - 3 $$).

### Step 3: Identify the solution set

The solution set of the system of inequalities is the region where the shaded areas from both inequalities overlap.

### Summary of the Graph

- The line $$ y = -\frac{1}{2}x + 2 $$ is drawn with a solid line (since the inequality is $$ \leq $$).
- The parabola $$ y = x^2 - 3 $$ is drawn with a solid line (since the inequality is $$ \geq $$).
- The area below the line and above the parabola represents the solution set.

### Final Note

To visualize this, you would typically use graphing software or graph paper to accurately plot the line and the parabola, ensuring to shade the correct regions. The intersection points of the line and the parabola can also be found to determine the boundaries of the solution set more precisely.
The solution set is the region where the line $$ y = -\frac{1}{2}x + 2 $$ and the parabola $$ y = x^2 - 3 $$ overlap.

Graph the solution set of the following system of inequalities.

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Solución

Step 1: Graph the first inequality

Step 2: Graph the second inequality

Step 3: Identify the solution set

Summary of the Graph

Final Note

Bonus Knowledge

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