Graph the solutions of the given system of linear inequalities. $$ \left\{\begin{array}{l}y

Question

Graph the solutions of the given system of linear inequalities. $$ \left\{\begin{array}{l}y<3 x-3 \ y \leq x+2\end{array}ight. $$

UpStudy AI · Accepted Answer

To graph the solutions of the given system of linear inequalities:

$$
\left\{
\begin{array}{l}
y < 3x - 3 \
y \leq x + 2
\end{array}
\right.
$$

follow these steps:

### 1. Graph the Boundary Lines

Each inequality corresponds to a boundary line:

1. **For $$ y < 3x - 3 $$:**
   - **Boundary Line:** $$ y = 3x - 3 $$
   - **Type of Line:** **Dashed** (since the inequality is strict `<`, the line is not included in the solution).
   
2. **For $$ y \leq x + 2 $$:**
   - **Boundary Line:** $$ y = x + 2 $$
   - **Type of Line:** **Solid** (since the inequality includes `≤`, the line is part of the solution).

### 2. Find the Intersection Point

To determine where the two lines intersect, solve the system:

$$
\begin{cases}
y = 3x - 3 \
y = x + 2
\end{cases}
$$

Set the equations equal to each other:

$$
3x - 3 = x + 2 \
\Rightarrow 2x = 5 \
\Rightarrow x = \frac{5}{2}
$$

Substitute $$ x = \frac{5}{2} $$ into $$ y = x + 2 $$:

$$
y = \frac{5}{2} + 2 = \frac{9}{2}
$$

**Intersection Point:** $$ \left( \frac{5}{2}, \frac{9}{2} \right) $$ or $$ (2.5, 4.5) $$

### 3. Determine the Shaded Regions

- **For $$ y < 3x - 3 $$:** Shade **below** the dashed line $$ y = 3x - 3 $$.
- **For $$ y \leq x + 2 $$:** Shade **below or on** the solid line $$ y = x + 2 $$.

### 4. Combined Solution

The solution to the system is the set of points that satisfy **both** inequalities simultaneously. This is the overlapping (intersecting) shaded region below both lines.

### 5. Visual Representation

While I cannot provide a visual graph here, you can visualize it as follows:

1. **Draw the lines:**
   - **Dashed Line:** $$ y = 3x - 3 $$ (e.g., passes through points like $$ (0, -3) $$ and $$ (1, 0) $$)
   - **Solid Line:** $$ y = x + 2 $$ (e.g., passes through points like $$ (0, 2) $$ and $$ (2, 4) $$)
   
2. **Shade the appropriate regions:**
   - Below the dashed line for $$ y < 3x - 3 $$.
   - Below the solid line for $$ y \leq x + 2 $$.
   
3. **Identify the overlapping area:** This is your solution set, consisting of all points that lie below **both** lines.

### 6. Example Points in the Solution

To confirm, let's test a point in the overlapping shaded region, such as $$ (1, 1) $$:

- **First Inequality:** $$ 1 < 3(1) - 3 $$ → $$ 1 < 0 $$ → **False**
  
Oops! This point does **not** satisfy both inequalities. Let's try $$ (0, 0) $$:

- **First Inequality:** $$ 0 < 3(0) - 3 $$ → $$ 0 < -3 $$ → **False**
  
Another try: $$ (2, 3) $$:

- **First Inequality:** $$ 3 < 3(2) - 3 $$ → $$ 3 < 3 $$ → **False**
  
It seems earlier assumptions about shading might need correction. Let's reassess:

### Correct Shading Approach

Upon closer inspection, for the inequality $$ y < 3x - 3 $$, the region **above** the dashed line is **not** correct. The inequality $$ y < 3x - 3 $$ actually represents the region **below** the line. However, our initial test points did not lie in the correct overlapping region.

Let's accurately identify the overlapping region by choosing points that satisfy both inequalities:

- Choose $$ (3, 5) $$:
  - **First Inequality:** $$ 5 < 3(3) - 3 $$ → $$ 5 < 6 $$ → **True**
  - **Second Inequality:** $$ 5 \leq 3 + 2 $$ → $$ 5 \leq 5 $$ → **True**
  
- Choose $$ (4, 6) $$:
  - **First Inequality:** $$ 6 < 3(4) - 3 $$ → $$ 6 < 9 $$ → **True**
  - **Second Inequality:** $$ 6 \leq 4 + 2 $$ → $$ 6 \leq 6 $$ → **True**

Thus, the correct overlapping shaded region is **below** both lines. Ensure to test points carefully to confirm the correct shading.

### Final Notes

When graphing multiple inequalities:

1. **Graph each boundary line first**, using solid or dashed lines based on the inequality type.
2. **Shade the appropriate side** of each line according to the inequality.
3. **Identify the overlapping shaded region**, which represents the solution to the system.

I hope this helps you visualize and graph the solution to the system of inequalities!
To graph the solutions of the system of inequalities: $$ \left\{ \begin{array}{l} y < 3x - 3 \ y \leq x + 2 \end{array} \right. $$ 1. **Graph the boundary lines**: - **Dashed line**: $$ y = 3x - 3 $$ - **Solid line**: $$ y = x + 2 $$ 2. **Find the intersection point** at $$ \left( \frac{5}{2}, \frac{9}{2} \right) $$ or $$ (2.5, 4.5) $$. 3. **Shade the regions**: - Below the dashed line for $$ y < 3x - 3 $$ - Below or on the solid line for $$ y \leq x + 2 $$ 4. **The overlapping shaded area** represents the solution to the system. Ensure to test points to confirm the correct shading.

Graph the solutions of the given system of linear inequalities. \( \left\{\begin{array}{l}y<3 x-3 \\ y \leq x+2\end{array}\right. \)

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