#3 ### Inequalities and Graphing Relations**3. Solve the following inequalities and represent the solution in interval notation. Show your work!****(a) $6x^2 - 4 \geq 5x + 2$****(b) $x|x - 4| < 3$****4. Sketch the graph of the relation $R = \{(x, y) \mid x^2 - 2x - 2 \leq y \leq 2x - x^2 + 2\}$.**### Instructions for Students:1. **Solving Inequalities:** - For each inequality, isolate the variable $x$ and determine the solution set. - Convert the solution set into interval notation, which is a way of writing subsets of the real number line.2. **Graphing the Relation:** - Identify the boundaries defined by the inequalities. - Draw the graphs of these boundary equations on a coordinate plane. - Shade the region that represents the solution to the inequalities.### Detailed Graph Explanation:In problem 4, you will be plotting two quadratic functions to determine the region where the inequalities hold:- The first boundary is $y = x^2 - 2x - 2$. - This is a parabola opening upwards.- The second boundary is $y = 2x - x^2 + 2$. - This is a parabola opening downwards.### Steps to Graph:1. **Find Vertex and Intercepts:** - Determine the vertex of each parabola. - Find the x-intercepts by solving $x^2 - 2x - 2 = 0$ and $2x - x^2 + 2 = 0$.2. **Plot Points:** - Plot the vertex and intercept points on a graph. - Draw the parabolas using these points.3. **Shading the Region:** - Identify the region between these two curves. - Shade the area where the inequality $x^2 - 2x - 2 \leq y \leq 2x - x^2 + 2$ holds.By following these steps, you will understand how to solve inequalities and represent them graphically, a crucial skill in algebra and calculus.

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Description: #3 ### Inequalities and Graphing Relations**3. Solve the following inequalities and represent the solution in interval notation. Show your work!****(a) $6x^2 - 4 \geq 5x + 2$****(b) $x|x - 4| < 3$****4. Sketch the graph of the relation \(R = \{(x

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Answered: 3. Solve the following inequalities and…

Math

Algebra

3. Solve the following inequalities and represent the solution in interval notation. Show your work! (a) 6x² – 4 > 5x + 2

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Inequalities and Graphing Relations

**3. Solve the following inequalities and represent the solution in interval notation. Show your work!**

**(a) $6x^2 - 4 \geq 5x + 2$**

**(b) $x|x - 4| < 3$**

**4. Sketch the graph of the relation $R = \{(x, y) \mid x^2 - 2x - 2 \leq y \leq 2x - x^2 + 2\}$.**

### Instructions for Students:

1. **Solving Inequalities:**
- For each inequality, isolate the variable $x$ and determine the solution set.
- Convert the solution set into interval notation, which is a way of writing subsets of the real number line.

2. **Graphing the Relation:**
- Identify the boundaries defined by the inequalities.
- Draw the graphs of these boundary equations on a coordinate plane.
- Shade the region that represents the solution to the inequalities.

### Detailed Graph Explanation:

In problem 4, you will be plotting two quadratic functions to determine the region where the inequalities hold:
- The first boundary is $y = x^2 - 2x - 2$.
- This is a parabola opening upwards.
- The second boundary is $y = 2x - x^2 + 2$.
- This is a parabola opening downwards.

### Steps to Graph:
1. **Find Vertex and Intercepts:**
- Determine the vertex of each parabola.
- Find the x-intercepts by solving $x^2 - 2x - 2 = 0$ and $2x - x^2 + 2 = 0$.

2. **Plot Points:**
- Plot the vertex and intercept points on a graph.
- Draw the parabolas using these points.

3. **Shading the Region:**
- Identify the region between these two curves.
- Shade the area where the inequality $x^2 - 2x - 2 \leq y \leq 2x - x^2 + 2$ holds.

By following these steps, you will understand how to solve inequalities and represent them graphically, a crucial skill in algebra and calculus.

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