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. $$ \left\{\begin{array}{l}y<5 x-5 \ y \leq x+2\end{array}ight. $$

UpStudy AI · Accepted Answer

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

$$
\begin{cases}
y < 5x - 5 \
y \leq x + 2
\end{cases}
$$

follow these step-by-step instructions:

---

### **1. Graph the Boundary Lines**

First, treat each inequality as an equation to find its boundary line.

#### **a. $$ y = 5x - 5 $$**

- **Slope-Intercept Form:** This equation is already in the form $$ y = mx + b $$, where:
  - **Slope (m):** 5
  - **Y-intercept (b):** -5

- **Plotting Points:**
  - **When $$ x = 0 $$:** $$ y = 5(0) - 5 = -5 $$ → Point (0, -5)
  - **When $$ x = 1 $$:** $$ y = 5(1) - 5 = 0 $$ → Point (1, 0)
  
- **Draw the Line:** Plot the points (0, -5) and (1, 0) and draw a straight line through them.

- **Line Style:** Since the inequality is **$$ y < 5x - 5 $$** (a strict inequality), use a **dashed line**. This indicates that points on the line are **not** included in the solution.

---

#### **b. $$ y = x + 2 $$**

- **Slope-Intercept Form:** This equation is also in the form $$ y = mx + b $$, where:
  - **Slope (m):** 1
  - **Y-intercept (b):** 2

- **Plotting Points:**
  - **When $$ x = 0 $$:** $$ y = 0 + 2 = 2 $$ → Point (0, 2)
  - **When $$ x = 2 $$:** $$ y = 2 + 2 = 4 $$ → Point (2, 4)
  
- **Draw the Line:** Plot the points (0, 2) and (2, 4) and draw a straight line through them.

- **Line Style:** Since the inequality is **$$ y \leq x + 2 $$** (a non-strict inequality), use a **solid line**. This indicates that points on the line **are** included in the solution.

---

### **2. Determine the Shaded Regions**

Next, decide which side of each boundary line to shade based on the inequalities.

#### **a. $$ y < 5x - 5 $$**

- **Test a Point:** Choose an easy point not on the line, such as (0,0).
  - Substitute into the inequality: $$ 0 < 5(0) - 5 $$ → $$ 0 < -5 $$ → **False**
- **Shading Direction:** Since (0,0) does not satisfy the inequality, shade the **opposite side** of the line from (0,0). This means shading **below** the dashed line $$ y = 5x - 5 $$.

#### **b. $$ y \leq x + 2 $$**

- **Test a Point:** Again, use (0,0).
  - Substitute into the inequality: $$ 0 \leq 0 + 2 $$ → $$ 0 \leq 2 $$ → **True**
- **Shading Direction:** Since (0,0) satisfies the inequality, shade the **same side** of the line as (0,0). This means shading **below** the solid line $$ y = x + 2 $$.

---

### **3. Identify the Solution Region**

The solution to the system of inequalities is the **overlapping shaded area** that satisfies **both** inequalities simultaneously.

- **Boundaries:**
  - **Below the dashed line $$ y = 5x - 5 $$**
  - **Below the solid line $$ y \leq x + 2 $$**

- **Intersection Point:** It's often helpful to find where the two boundary lines intersect to better understand the solution region.
  
  - **Set the equations equal to each other:**
    $$
    5x - 5 = x + 2
    $$
  
  - **Solve for $$ x $$:**
    $$
    5x - x = 2 + 5 \
    4x = 7 \
    x = \frac{7}{4}
    $$
  
  - **Find $$ y $$:**
    $$
    y = \frac{7}{4} + 2 = \frac{15}{4}
    $$
  
  - **Intersection Point:** $$ \left( \frac{7}{4}, \frac{15}{4} \right) $$

This point is where the two boundary lines meet. The solution region lies **below both lines**, forming a region below the dashed line $$ y = 5x - 5 $$ and below the solid line $$ y = x + 2 $$.

---

### **4. Final Graph Overview**

- **Dashed Line:** $$ y = 5x - 5 $$ with shading below.
- **Solid Line:** $$ y = x + 2 $$ with shading below.
- **Solution Region:** The area where the two shaded regions overlap, lying below both lines.

Here's a simplified sketch to visualize the graph:

```
y
|
|                     *
|                   * |
|                 *   |
|               *     |
|-------------*-------*----- x
|           * |      /
|         *   |    /
|       *     |  /
|     *       |/
|
```

*(Note: The "*" represents the intersection point at $$ \left( \frac{7}{4}, \frac{15}{4} \right) $$. The actual graph would show two lines crossing at this point with the appropriate shading below both lines.)*

---

### **Summary**

- **Draw both boundary lines** ($$ y = 5x - 5 $$ as dashed and $$ y = x + 2 $$ as solid).
- **Shade below** both lines.
- **The overlapping shaded area** represents all the points $$ (x, y) $$ that satisfy both inequalities simultaneously.

By following these steps, you can accurately graph the solution set for the system of linear inequalities.
To graph the solutions of the system of inequalities: $$ \begin{cases} y < 5x - 5 \ y \leq x + 2 \end{cases} $$ 1. **Graph the boundary lines**: - $$ y = 5x - 5 $$ (dashed line) - $$ y = x + 2 $$ (solid line) 2. **Determine the shaded regions**: - Shade below $$ y = 5x - 5 $$ - Shade below $$ y = x + 2 $$ 3. **Identify the solution region**: - The overlapping shaded area where both inequalities are satisfied. The solution is the area below both lines, including the line $$ y = x + 2 $$ and excluding the line $$ y = 5x - 5 $$.

Graph the solutions of the given system of linear inequalities. \( \left\{\begin{array}{l}y<5 x-5 \\ y \leq x+2\end{array}\right. \)

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