Question
Solve the absolute value inequality.
StartAbsoluteValue 4 left parenthesis x minus 1 right parenthesis plus 6 EndAbsoluteValue less than or equals 14
Question content area bottom
Part 1
Rewrite the inequality without absolute value bars. Select the correct choice below and fill in the answer box(es) to complete your choice.
(Simplify your answers.)
A.enter your response here less than or equals 4 left parenthesis x minus 1 right parenthesis plus 6 less than or equals enter your response here
enter your response hereless than or equals4 left parenthesis x minus 1 right parenthesis plus 6less than or equals
enter your response here
B.4 left parenthesis x minus 1 right parenthesis plus 6 less than or equals enter your response here
4 left parenthesis x minus 1 right parenthesis plus 6less than or equals
enter your response here
C.
4 left parenthesis x minus 1 right parenthesis plus 6less than or equals
enter your response here or 4 left parenthesis x minus 1 right parenthesis plus 6greater than or equals
enter your response here
Part 2
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The solution set in interval notation is
enter your response here.
(Simplify your answer.)
B.
The solution set is empty set.
StartAbsoluteValue 4 left parenthesis x minus 1 right parenthesis plus 6 EndAbsoluteValue less than or equals 14
Question content area bottom
Part 1
Rewrite the inequality without absolute value bars. Select the correct choice below and fill in the answer box(es) to complete your choice.
(Simplify your answers.)
A.enter your response here less than or equals 4 left parenthesis x minus 1 right parenthesis plus 6 less than or equals enter your response here
enter your response hereless than or equals4 left parenthesis x minus 1 right parenthesis plus 6less than or equals
enter your response here
B.4 left parenthesis x minus 1 right parenthesis plus 6 less than or equals enter your response here
4 left parenthesis x minus 1 right parenthesis plus 6less than or equals
enter your response here
C.
4 left parenthesis x minus 1 right parenthesis plus 6less than or equals
enter your response here or 4 left parenthesis x minus 1 right parenthesis plus 6greater than or equals
enter your response here
Part 2
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The solution set in interval notation is
enter your response here.
(Simplify your answer.)
B.
The solution set is empty set.
Answers
Answered by
GPT-4o mini
To solve the absolute value inequality \( |4(x - 1) + 6| \leq 14 \), we first rewrite it without absolute value bars.
We know that the expression within absolute value can be either negative or positive. Therefore, we can write the inequality as two separate inequalities:
\[
-14 \leq 4(x - 1) + 6 \leq 14
\]
### Part 1
Let's break this down into two parts.
1. For the left side of the inequality:
\[
4(x - 1) + 6 \geq -14
\]
Simplifying this, we subtract 6 from both sides:
\[
4(x - 1) \geq -20
\]
Now, divide by 4:
\[
x - 1 \geq -5 \quad \Rightarrow \quad x \geq -4
\]
2. For the right side of the inequality:
\[
4(x - 1) + 6 \leq 14
\]
Again, subtract 6 from both sides:
\[
4(x - 1) \leq 8
\]
Divide by 4:
\[
x - 1 \leq 2 \quad \Rightarrow \quad x \leq 3
\]
So our rewritten inequality without absolute value bars is:
\[
-4 \leq x \leq 3
\]
Hence, selecting the option from Part 1:
**Answer A:**
\[
-4 \text{ less than or equal to } 4(x - 1) + 6 \text{ less than or equal to } 14
\]
### Part 2
Next, we find the solution set.
The solution we derived shows that \( x \) is bounded between \( -4 \) and \( 3 \). In interval notation, this is expressed as:
\[
[-4, 3]
\]
So, selecting the option from Part 2:
**Answer A:**
The solution set in interval notation is \( [-4, 3] \).
We know that the expression within absolute value can be either negative or positive. Therefore, we can write the inequality as two separate inequalities:
\[
-14 \leq 4(x - 1) + 6 \leq 14
\]
### Part 1
Let's break this down into two parts.
1. For the left side of the inequality:
\[
4(x - 1) + 6 \geq -14
\]
Simplifying this, we subtract 6 from both sides:
\[
4(x - 1) \geq -20
\]
Now, divide by 4:
\[
x - 1 \geq -5 \quad \Rightarrow \quad x \geq -4
\]
2. For the right side of the inequality:
\[
4(x - 1) + 6 \leq 14
\]
Again, subtract 6 from both sides:
\[
4(x - 1) \leq 8
\]
Divide by 4:
\[
x - 1 \leq 2 \quad \Rightarrow \quad x \leq 3
\]
So our rewritten inequality without absolute value bars is:
\[
-4 \leq x \leq 3
\]
Hence, selecting the option from Part 1:
**Answer A:**
\[
-4 \text{ less than or equal to } 4(x - 1) + 6 \text{ less than or equal to } 14
\]
### Part 2
Next, we find the solution set.
The solution we derived shows that \( x \) is bounded between \( -4 \) and \( 3 \). In interval notation, this is expressed as:
\[
[-4, 3]
\]
So, selecting the option from Part 2:
**Answer A:**
The solution set in interval notation is \( [-4, 3] \).