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 7/30**
---
**Graph the solution to** \(| -3a + 3 | - 6 < 12\)
---
**Graph Description:**
The image shows a number line ranging from -8 to 8 with points plotted on the line. There are several interactive elements, including options for "UNDO," "REDO," "RESET," and "TRASH." Below the number line, there are various circle and arrow symbols that likely represent different states or types of points.
### Analysis and Explanation:
To solve the inequality \(| -3a + 3 | - 6 < 12\):
1. **Step 1:** Isolate the absolute value expression:
\[
| -3a + 3 | < 18
\]
2. **Step 2:** Consider the definition of absolute value:
\[
-18 < -3a + 3 < 18
\]
3. **Step 3:** Solve the compound inequality:
- For the left part:
\[
-18 < -3a + 3 \implies -21 < -3a \implies 7 > a \implies a < 7
\]
- For the right part:
\[
-3a + 3 < 18 \implies -3a < 15 \implies a > -5
\]
Putting it all together:
\[
-5 < a < 7
\]
### Graph Representation:
On the number line, plot open circles (indicating non-inclusive values) at -5 and 7, and shade the section between these two points to denote all values in this range.
In summary, the solution set for \(| -3a + 3 | - 6 < 12\) is graphed as \( -5 < a < 7 \) on a number line from -8 to 8 with points and shaded regions indicating the range.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F747abddb-1b1f-4ced-a4f8-74d5c7293d32%2F578d5d9a-d131-4e1a-8095-d4bcc6762b53%2Fymir20s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 7/30**
---
**Graph the solution to** \(| -3a + 3 | - 6 < 12\)
---
**Graph Description:**
The image shows a number line ranging from -8 to 8 with points plotted on the line. There are several interactive elements, including options for "UNDO," "REDO," "RESET," and "TRASH." Below the number line, there are various circle and arrow symbols that likely represent different states or types of points.
### Analysis and Explanation:
To solve the inequality \(| -3a + 3 | - 6 < 12\):
1. **Step 1:** Isolate the absolute value expression:
\[
| -3a + 3 | < 18
\]
2. **Step 2:** Consider the definition of absolute value:
\[
-18 < -3a + 3 < 18
\]
3. **Step 3:** Solve the compound inequality:
- For the left part:
\[
-18 < -3a + 3 \implies -21 < -3a \implies 7 > a \implies a < 7
\]
- For the right part:
\[
-3a + 3 < 18 \implies -3a < 15 \implies a > -5
\]
Putting it all together:
\[
-5 < a < 7
\]
### Graph Representation:
On the number line, plot open circles (indicating non-inclusive values) at -5 and 7, and shade the section between these two points to denote all values in this range.
In summary, the solution set for \(| -3a + 3 | - 6 < 12\) is graphed as \( -5 < a < 7 \) on a number line from -8 to 8 with points and shaded regions indicating the range.
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