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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![# Solving Inequalities and Representation in Interval Notation
## Problem 3
### Instructions
Solve the following inequalities and represent the solution in interval notation. Show your work!
### Given Inequalities
**(a)** \( 6x^2 - 4 \geq 5x + 2 \)
**(b)** \( x|x - 4| < 3 \)
### Solution
#### Part (a): Quadratic Inequality
To solve the inequality \( 6x^2 - 4 \geq 5x + 2 \):
1. Rewrite the inequality in standard form.
\[
6x^2 - 4 \geq 5x + 2 \implies 6x^2 - 5x - 6 \geq 0
\]
2. Factor the quadratic expression on the left-hand side (if possible) or use the quadratic formula to find the critical points.
3. Determine the intervals to test for the inequality.
4. Represent the solution in interval notation.
#### Part (b): Absolute Value Inequality
To solve the inequality \( x|x - 4| < 3 \):
1. Consider the definition of absolute value and break the problem into cases:
- Case 1: \( x \geq 4 \) (where \( |x - 4| = x - 4 \))
- Case 2: \( x < 4 \) (where \( |x - 4| = 4 - x \))
2. Solve each case separately for \( x \cdot (x - 4) < 3 \) and \( x \cdot (4 - x) < 3 \).
3. Combine the results from both cases and represent the solution in interval notation.
### Graphical Representations
The graphical representation of the solutions (if applicable) provides a visual aid to understanding the intervals where these inequalities hold true.
For further details, including step-by-step solutions, refer to the respective sections on solving quadratic and absolute value inequalities.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2ea4237-947f-4fb7-b8d4-fb858663c4d1%2F6ecedbd2-9117-4942-b283-cd51b56fe68d%2Fdvvw34q.jpeg&w=3840&q=75)
Transcribed Image Text:# Solving Inequalities and Representation in Interval Notation
## Problem 3
### Instructions
Solve the following inequalities and represent the solution in interval notation. Show your work!
### Given Inequalities
**(a)** \( 6x^2 - 4 \geq 5x + 2 \)
**(b)** \( x|x - 4| < 3 \)
### Solution
#### Part (a): Quadratic Inequality
To solve the inequality \( 6x^2 - 4 \geq 5x + 2 \):
1. Rewrite the inequality in standard form.
\[
6x^2 - 4 \geq 5x + 2 \implies 6x^2 - 5x - 6 \geq 0
\]
2. Factor the quadratic expression on the left-hand side (if possible) or use the quadratic formula to find the critical points.
3. Determine the intervals to test for the inequality.
4. Represent the solution in interval notation.
#### Part (b): Absolute Value Inequality
To solve the inequality \( x|x - 4| < 3 \):
1. Consider the definition of absolute value and break the problem into cases:
- Case 1: \( x \geq 4 \) (where \( |x - 4| = x - 4 \))
- Case 2: \( x < 4 \) (where \( |x - 4| = 4 - x \))
2. Solve each case separately for \( x \cdot (x - 4) < 3 \) and \( x \cdot (4 - x) < 3 \).
3. Combine the results from both cases and represent the solution in interval notation.
### Graphical Representations
The graphical representation of the solutions (if applicable) provides a visual aid to understanding the intervals where these inequalities hold true.
For further details, including step-by-step solutions, refer to the respective sections on solving quadratic and absolute value inequalities.
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