Question
Solve |2x|<18 . Select the interval notation that represents the solution.(1 point)
Answers
Answer
Write the solution to −9+4|x−1|≤7 as a compound inequality.(1 point)
Answer
Solve the absolute value inequality |0.5x+9|≤−2 .(1 point)
Answer
Question When solving the absolute value inequality 2|x+5|<8 , it is necessary to first isolate the absolute value. Which absolute value rule must be applied, or which special case is this an example of?(1 point) Responses
Answer
Question When solving the absolute value inequality 2|x+5|<8 , it is necessary to first isolate the absolute value. Which absolute value rule must be applied, or which special case is this an example of?(1 point) Responses
A No solution
B All real numbers are solutions.
C "and rule"
D "or rule"
A No solution
B All real numbers are solutions.
C "and rule"
D "or rule"
Answer
Solve the absolute value inequality 3|7x|+8≤6 .(1 point)
Answered by
GPT-4o mini
To solve the inequality \( |2x| < 18 \), we first recognize what the absolute value means. The inequality \( |2x| < 18 \) can be rewritten as two separate inequalities:
\[
-18 < 2x < 18
\]
Now, we can solve each part of the compound inequality.
1. For the left side:
\[
-18 < 2x
\]
Dividing both sides by 2 gives:
\[
-9 < x
\]
or
\[
x > -9
\]
2. For the right side:
\[
2x < 18
\]
Dividing both sides by 2 gives:
\[
x < 9
\]
Combining both results, we have:
\[
-9 < x < 9
\]
In interval notation, this is written as:
\[
(-9, 9)
\]
Thus, the solution in interval notation is \((-9, 9)\).
\[
-18 < 2x < 18
\]
Now, we can solve each part of the compound inequality.
1. For the left side:
\[
-18 < 2x
\]
Dividing both sides by 2 gives:
\[
-9 < x
\]
or
\[
x > -9
\]
2. For the right side:
\[
2x < 18
\]
Dividing both sides by 2 gives:
\[
x < 9
\]
Combining both results, we have:
\[
-9 < x < 9
\]
In interval notation, this is written as:
\[
(-9, 9)
\]
Thus, the solution in interval notation is \((-9, 9)\).
Answered by
GPT-4o mini
To solve the inequality \( -9 + 4|x - 1| \leq 7 \), we will first isolate the absolute value expression.
1. **Add 9 to both sides**:
\[
4|x - 1| \leq 7 + 9
\]
\[
4|x - 1| \leq 16
\]
2. **Divide both sides by 4**:
\[
|x - 1| \leq 4
\]
Now, we can rewrite the absolute value inequality as a compound inequality:
\[
-4 \leq x - 1 \leq 4
\]
3. **Add 1 to all parts of the inequality**:
\[
-4 + 1 \leq x - 1 + 1 \leq 4 + 1
\]
\[
-3 \leq x \leq 5
\]
Thus, the solution in the form of a compound inequality is:
\[
-3 \leq x \leq 5
\]
1. **Add 9 to both sides**:
\[
4|x - 1| \leq 7 + 9
\]
\[
4|x - 1| \leq 16
\]
2. **Divide both sides by 4**:
\[
|x - 1| \leq 4
\]
Now, we can rewrite the absolute value inequality as a compound inequality:
\[
-4 \leq x - 1 \leq 4
\]
3. **Add 1 to all parts of the inequality**:
\[
-4 + 1 \leq x - 1 + 1 \leq 4 + 1
\]
\[
-3 \leq x \leq 5
\]
Thus, the solution in the form of a compound inequality is:
\[
-3 \leq x \leq 5
\]
Answered by
GPT-4o mini
To solve the absolute value inequality \( |0.5x + 9| \leq -2 \), we need to consider the properties of absolute values.
The absolute value of any expression is always non-negative, which means it cannot be less than or equal to a negative number. Therefore, the inequality \( |0.5x + 9| \leq -2 \) has **no solution.**
In conclusion, the solution set is empty, and we can denote it as:
\[
\text{No solution}
\] or
\[
\emptyset
\]
The absolute value of any expression is always non-negative, which means it cannot be less than or equal to a negative number. Therefore, the inequality \( |0.5x + 9| \leq -2 \) has **no solution.**
In conclusion, the solution set is empty, and we can denote it as:
\[
\text{No solution}
\] or
\[
\emptyset
\]
Answered by
GPT-4o mini
To solve the absolute value inequality \( 2|x + 5| < 8 \), you will first need to isolate the absolute value.
1. **Isolation**: Divide both sides of the inequality by 2:
\[
|x + 5| < 4
\]
Now that we have isolated the absolute value, we can apply the property of absolute values.
### Absolute Value Rule
The absolute value inequality \( |A| < B \) can be transformed into a compound inequality:
\[
-B < A < B
\]
This means we can express \( |x + 5| < 4 \) as:
\[
-4 < x + 5 < 4
\]
### Special Case
In this context, this is an example of the rule that states:
- For any constant \( B > 0 \), \( |A| < B \) leads to \( -B < A < B \).
In summary, the necessary rule to apply is that when the absolute value is less than a positive number, you can create a compound inequality that encompasses that range, indicating that the expression inside the absolute value lies within that range.
1. **Isolation**: Divide both sides of the inequality by 2:
\[
|x + 5| < 4
\]
Now that we have isolated the absolute value, we can apply the property of absolute values.
### Absolute Value Rule
The absolute value inequality \( |A| < B \) can be transformed into a compound inequality:
\[
-B < A < B
\]
This means we can express \( |x + 5| < 4 \) as:
\[
-4 < x + 5 < 4
\]
### Special Case
In this context, this is an example of the rule that states:
- For any constant \( B > 0 \), \( |A| < B \) leads to \( -B < A < B \).
In summary, the necessary rule to apply is that when the absolute value is less than a positive number, you can create a compound inequality that encompasses that range, indicating that the expression inside the absolute value lies within that range.
Answered by
GPT-4o mini
When solving the absolute value inequality \( 2|x + 5| < 8 \), after isolating the absolute value, you can express it as \( |x + 5| < 4 \). This leads to a compound inequality which represents the values of \( x \) that satisfy the original inequality.
In this context, when solving inequalities of the form \( |A| < B \), you're applying the **"and rule,"** which means that both parts of the resulting compound inequality must be satisfied simultaneously.
Therefore, the correct response is:
**C "and rule"**
In this context, when solving inequalities of the form \( |A| < B \), you're applying the **"and rule,"** which means that both parts of the resulting compound inequality must be satisfied simultaneously.
Therefore, the correct response is:
**C "and rule"**
Answered by
GPT-4o mini
To solve the absolute value inequality \( 3|7x| + 8 \leq 6 \), we first need to isolate the absolute value term.
1. **Subtract 8 from both sides**:
\[
3|7x| \leq 6 - 8
\]
\[
3|7x| \leq -2
\]
2. **Divide both sides by 3**:
\[
|7x| \leq -\frac{2}{3}
\]
Since the absolute value of any expression is always non-negative (i.e., \( |A| \geq 0 \)), the expression \( |7x| \leq -\frac{2}{3} \) cannot hold true because it is set to be less than a negative number.
### Conclusion
Thus, there is **no solution** to the inequality \( 3|7x| + 8 \leq 6 \).
We can denote the solution set as:
\[
\text{No solution} \quad \text{or} \quad \emptyset
\]
1. **Subtract 8 from both sides**:
\[
3|7x| \leq 6 - 8
\]
\[
3|7x| \leq -2
\]
2. **Divide both sides by 3**:
\[
|7x| \leq -\frac{2}{3}
\]
Since the absolute value of any expression is always non-negative (i.e., \( |A| \geq 0 \)), the expression \( |7x| \leq -\frac{2}{3} \) cannot hold true because it is set to be less than a negative number.
### Conclusion
Thus, there is **no solution** to the inequality \( 3|7x| + 8 \leq 6 \).
We can denote the solution set as:
\[
\text{No solution} \quad \text{or} \quad \emptyset
\]
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