### Solving Inequalities Using Graphs#### Inequality to Solve:We need to solve the inequality $ R(x) \leq 0 $, where:\[ R(x) = \frac{3x + 3}{2x^2 + 4} \]#### Given Function Graph:The graph of the function $ R(x) = \frac{3x + 3}{2x^2 + 4} $ is displayed below.#### Graph Analysis:1. **Axes and Quadrants**: - The horizontal axis (x-axis) ranges from approximately -8 to 4. - The vertical axis (y-axis) ranges from approximately -6 to 8.2. **Asymptotes**: - Vertical Asymptote: $ x = -2 $. This is represented by a dashed red vertical line at $ x = -2 $. - Horizontal Asymptote: $ y = \frac{3}{2} $. This is represented by a dashed red horizontal line at $ y = \frac{3}{2} $.3. **Graph Behavior**: - For $ x $ values less than the vertical asymptote (-2), the curve starts from negative infinity, approaches the vertical asymptote, and then continues increasing without crossing it. - For $ x $ values greater than the vertical asymptote (-2), the curve starts from positive infinity and approaches the horizontal asymptote, not crossing it. #### Solving the Inequality:We want to find the values of $ x $ for which $ R(x) \leq 0 $.- Observing the graph, the function $ R(x) $ is below the x-axis (negative values) when: - $ x < -2 $#### Solution Set:The solution set for $ R(x) \leq 0 $ is:\[ (-\infty, -2) \]Please express your answer in interval notation.**Box to input answer**: (Type your answer in interval notation.)

The function is R(x)=3x-32x+4. The given graph is

Answered: 3x +3 Solve the inequality R(x) s0,…

3x +3 Solve the inequality R(x) s0, where R(x) = 4, by using the graph of the function. The solution set for R(x) S0 is. (Type your answer in interval notation.)

Intermediate Algebra

10th Edition

ISBN:9781285195728

Author:Jerome E. Kaufmann, Karen L. Schwitters

Publisher:Jerome E. Kaufmann, Karen L. Schwitters

Chapter9: Functions

Section9.1: Relations And Functions

Problem 75PS

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Question

### Solving Inequalities Using Graphs

#### Inequality to Solve:
We need to solve the inequality $ R(x) \leq 0 $, where:
\[ R(x) = \frac{3x + 3}{2x^2 + 4} \]

#### Given Function Graph:
The graph of the function $ R(x) = \frac{3x + 3}{2x^2 + 4} $ is displayed below.

#### Graph Analysis:

1. **Axes and Quadrants**:
- The horizontal axis (x-axis) ranges from approximately -8 to 4.
- The vertical axis (y-axis) ranges from approximately -6 to 8.

2. **Asymptotes**:
- Vertical Asymptote: $ x = -2 $. This is represented by a dashed red vertical line at $ x = -2 $.
- Horizontal Asymptote: $ y = \frac{3}{2} $. This is represented by a dashed red horizontal line at $ y = \frac{3}{2} $.

3. **Graph Behavior**:
- For $ x $ values less than the vertical asymptote (-2), the curve starts from negative infinity, approaches the vertical asymptote, and then continues increasing without crossing it.
- For $ x $ values greater than the vertical asymptote (-2), the curve starts from positive infinity and approaches the horizontal asymptote, not crossing it.

#### Solving the Inequality:

We want to find the values of $ x $ for which $ R(x) \leq 0 $.

- Observing the graph, the function $ R(x) $ is below the x-axis (negative values) when:
- $ x < -2 $

#### Solution Set:

The solution set for $ R(x) \leq 0 $ is:
\[ (-\infty, -2) \]

Please express your answer in interval notation.

**Box to input answer**: (Type your answer in interval notation.)

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