https://file.aiquickdraw.com/mathbot/file/1727449686257o7bfc74v.jpg

To describe the unshaded region in the diagram, we need to find the equations of the boundary lines and then determine the inequalities that correspond to the unshaded area.

### Step 1: Identify the equations of the boundary lines
- The **solid line** runs from \((-3, 0)\) to \((3, 6)\).
  - **Equation**: The slope \(m\) of the line is calculated as follows:
    \[
    m = \frac{6 - 0}{3 - (-3)} = \frac{6}{6} = 1
    \]
    Using the point-slope form \(y = mx + c\) and the point \((3, 6)\), we substitute to find the y-intercept \(c\):
    \[
    6 = 1(3) + c \Rightarrow c = 3
    \]
    So, the equation of the solid line is:
    \[
    y = x + 3
    \]

- The **dashed line** runs from \((-3, 3)\) to \((6, -3)\).
  - **Equation**: The slope \(m\) of this line is:
    \[
    m = \frac{-3 - 3}{6 - (-3)} = \frac{-6}{9} = -\frac{2}{3}
    \]
    Using the point \((6, -3)\) to find the y-intercept \(c\):
    \[
    -3 = -\frac{2}{3}(6) + c \Rightarrow -3 = -4 + c \Rightarrow c = 1
    \]
    So, the equation of the dashed line is:
    \[
    y = -\frac{2}{3}x + 1
    \]

### Step 2: Determine the inequalities for the unshaded region
- For the **solid line** \(y = x + 3\), the unshaded region is **below** this line, so the inequality is:
  \[
  y \leq x + 3
  \]

- For the **dashed line** \(y = -\frac{2}{3}x + 1\), the unshaded region is **above** this line, so the inequality is:
  \[
  y > -\frac{2}{3}x + 1
  \]
  Note that it's a strict inequality because the line is dashed.

### Final Answer:
The two inequalities that describe the unshaded region are:
\[
\boxed{y \leq x + 3 \quad \text{and} \quad y > -\frac{2}{3}x + 1}
\]

Do you want more details on this solution, or have any questions?

### Related Questions:
1. How do you determine whether to use a strict or non-strict inequality in a graph?
2. What is the point-slope form of a line, and how is it used to find the equation of a line?
3. How do you find the slope of a line given two points?
4. Can you explain why the inequalities are associated with the unshaded region rather than the shaded region?
5. What are the characteristics of the solution set for a system of linear inequalities?

**Tip:** Always pay attention to whether a line is dashed or solid when interpreting inequalities from a graph.

Write down the two inequalities that describe the unshaded region in the diagram below.

Solve a system of linear inequalities by graphing, with focus on finding the unshaded region between two lines. This problem covers the concepts of linear equations, inequalities, and graph interpretation, ideal for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation