#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…

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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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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

Topic Video

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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