Question
1. Misha found that the equation –|2x – 10| – 1 = 2 had two possible solutions: x = 3.5 and x = –6.5. Which explains whether or not her solutions are correct?
She is correct, because both solutions satisfy the equation.
She is not correct, because she made a sign error.
She is not correct, because there are no solutions.
She is not correct, because there is only one solution: x = 3.5.
2. Which equation has only one solution?
|x – 5| = –1
|–6 – 2x| = 8
|5x + 10| = 10
|–6x + 3| = 0
3. Which number line represents the solutions to |x + 4| = 2?
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Summary
The image contains four number lines, each ranging from -7 to 7. Blue dots are placed on specific numbers on each line.
First number line: Dots at -6 and -2.
Second number line: Dots at 2, 3, and 4.
Third number line: Dots at 2 and 6.
Fourth number line: Dots at -5 and -2.
4. Manuela solved the equation 3−2|0.5x+1.5|=2 for one solution. Her work is shown below.
3−2|0.5x+1.5|=2
−2|0.5x+1.5|=−1
|0.5x+1.5|=0.5
0.5x+1.5=0.5
0.5x=−1
x=−2
What is the other solution to the equation?
x=−6
x=−4
x=2
x=4
5. For water to be a liquid, its temperature must be within 50 Kelvin of 323 Kelvin. Which equation can be used to determine the minimum and maximum temperatures between which water is a liquid?
|323 – 50| = x
|323 + 50| = x
|x – 323| = 50
|x + 323| = 50
6. Morgan is walking her dog on an 8-meter-long leash. She is currently 500 meters from her house, so the maximum and minimum distances that the dog may be from the house can be found using the equation |x – 500| = 8. What are the minimum and maximum distances that Morgan’s dog may be from the house?
496 meters and 500 meters
500 meters and 508 meters
492 meters and 508 meters
496 meters and 504 meters
7. Which number line represents the solutions to |x – 5| = 1?
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Summary
The image contains four horizontal number lines ranging from -7 to 7. Each number line has blue dots indicating specific values. The positions of the blue dots on each number line are as follows:
-6
4 and 5
-3
-2
8. What is the solution to –2|2.2x – 3.3| = –6.6?
x = –3
x = 3
x = –3 or x = 0
x = 0 or x = 3
9. Which statement is true?
The equation –3|2x + 1.2| = –1 has no solution.
The equation 3.5|6x – 2| = 3.5 has one solution.
The equation 5|–3.1x + 6.9| = –3.5 has two solutions.
The equation –0.3|3 + 8x| = 0.9 has no solution.
10. Which number line represents the solutions to |–2x| = 4?
Image to Text
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10 -8 -6 -4 -2 0 2 4 6 8 10
Summary
The image contains four horizontal number lines, each ranging from -10 to 10. Each number line has a blue dot placed on specific points. The placement of the blue dots differs for each number line. The image seems to be part of a multiple-choice question, with each number line corresponding to an option.
She is correct, because both solutions satisfy the equation.
She is not correct, because she made a sign error.
She is not correct, because there are no solutions.
She is not correct, because there is only one solution: x = 3.5.
2. Which equation has only one solution?
|x – 5| = –1
|–6 – 2x| = 8
|5x + 10| = 10
|–6x + 3| = 0
3. Which number line represents the solutions to |x + 4| = 2?
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Summary
The image contains four number lines, each ranging from -7 to 7. Blue dots are placed on specific numbers on each line.
First number line: Dots at -6 and -2.
Second number line: Dots at 2, 3, and 4.
Third number line: Dots at 2 and 6.
Fourth number line: Dots at -5 and -2.
4. Manuela solved the equation 3−2|0.5x+1.5|=2 for one solution. Her work is shown below.
3−2|0.5x+1.5|=2
−2|0.5x+1.5|=−1
|0.5x+1.5|=0.5
0.5x+1.5=0.5
0.5x=−1
x=−2
What is the other solution to the equation?
x=−6
x=−4
x=2
x=4
5. For water to be a liquid, its temperature must be within 50 Kelvin of 323 Kelvin. Which equation can be used to determine the minimum and maximum temperatures between which water is a liquid?
|323 – 50| = x
|323 + 50| = x
|x – 323| = 50
|x + 323| = 50
6. Morgan is walking her dog on an 8-meter-long leash. She is currently 500 meters from her house, so the maximum and minimum distances that the dog may be from the house can be found using the equation |x – 500| = 8. What are the minimum and maximum distances that Morgan’s dog may be from the house?
496 meters and 500 meters
500 meters and 508 meters
492 meters and 508 meters
496 meters and 504 meters
7. Which number line represents the solutions to |x – 5| = 1?
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Summary
The image contains four horizontal number lines ranging from -7 to 7. Each number line has blue dots indicating specific values. The positions of the blue dots on each number line are as follows:
-6
4 and 5
-3
-2
8. What is the solution to –2|2.2x – 3.3| = –6.6?
x = –3
x = 3
x = –3 or x = 0
x = 0 or x = 3
9. Which statement is true?
The equation –3|2x + 1.2| = –1 has no solution.
The equation 3.5|6x – 2| = 3.5 has one solution.
The equation 5|–3.1x + 6.9| = –3.5 has two solutions.
The equation –0.3|3 + 8x| = 0.9 has no solution.
10. Which number line represents the solutions to |–2x| = 4?
Image to Text
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10 -8 -6 -4 -2 0 2 4 6 8 10
Summary
The image contains four horizontal number lines, each ranging from -10 to 10. Each number line has a blue dot placed on specific points. The placement of the blue dots differs for each number line. The image seems to be part of a multiple-choice question, with each number line corresponding to an option.
Answers
Answer
90 PERCENT thaNK YOU
Answered by
GPT-4o mini
Let's go through each question one by one to determine the correct answers.
1. **Misha found that the equation –|2x – 10| – 1 = 2 had two possible solutions: x = 3.5 and x = –6.5. Which explains whether or not her solutions are correct?**
To check Misha's solutions, we can substitute each value back into the original equation:
- For \( x = 3.5 \):
\[
-|2(3.5) - 10| - 1 = -|7 - 10| - 1 = -|-3| - 1 = -3 - 1 = -4 \neq 2
\]
- For \( x = -6.5 \):
\[
-|2(-6.5) - 10| - 1 = -|-13 - 10| - 1 = -|-23| - 1 = -23 - 1 = -24 \neq 2
\]
Both solutions are incorrect. Therefore, the answer is:
**She is not correct, because there are no solutions.**
2. **Which equation has only one solution?**
- \( |x – 5| = -1 \): No solution, as absolute values cannot equal a negative number.
- \( |–6 – 2x| = 8 \): Two solutions.
- \( |5x + 10| = 10 \): Two solutions.
- \( |–6x + 3| = 0 \): One solution (only when the expression inside the absolute value is 0).
Therefore, the answer is:
**|–6x + 3| = 0**
3. **Which number line represents the solutions to |x + 4| = 2?**
- The solutions to the equation are:
\[
x + 4 = 2 \Rightarrow x = -2 \quad \text{and} \quad x + 4 = -2 \Rightarrow x = -6
\]
- Thus, the solutions are \( x = -2 \) and \( x = -6 \).
The number line that shows dots at **-6 and -2** would be the correct one.
4. **Manuela solved the equation 3−2|0.5x+1.5|=2 for one solution. What is the other solution to the equation?**
- From her solution:
\[
|0.5x + 1.5| = 0.5
\]
- This will yield:
\[
0.5x + 1.5 = 0.5 \Rightarrow 0.5x = -1 \Rightarrow x = -2
\]
\[
0.5x + 1.5 = -0.5 \Rightarrow 0.5x = -2 \Rightarrow x = -4
\]
- Thus, the other solution is \( x = -4 \).
Therefore, the answer is:
**x = -4**
5. **For water to be a liquid, its temperature must be within 50 Kelvin of 323 Kelvin. Which equation can be used to determine the minimum and maximum temperatures?**
- The equation describes the distance from 323 to be ±50:
\[
|x - 323| = 50
\]
Therefore, the answer is:
**|x – 323| = 50**
6. **Morgan's dog on an 8-meter-long leash. What are the minimum and maximum distances?**
- The equation is \( |x - 500| = 8 \).
- This gives:
\[
x - 500 = 8 \Rightarrow x = 508
\]
\[
x - 500 = -8 \Rightarrow x = 492
\]
- Thus, the minimum distance is 492 meters and the maximum distance is 508 meters.
Therefore, the answer is:
**492 meters and 508 meters**
7. **Which number line represents the solutions to |x – 5| = 1?**
- The solutions are:
\[
x - 5 = 1 \Rightarrow x = 6
\]
\[
x - 5 = -1 \Rightarrow x = 4
\]
- Therefore, the number line showing dots at **4 and 6** would be the correct choice.
8. **What is the solution to –2|2.2x – 3.3| = –6.6?**
- Simplifying gives us:
\[
|2.2x - 3.3| = 3.3
\]
- This yields:
\[
2.2x - 3.3 = 3.3 \Rightarrow 2.2x = 6.6 \Rightarrow x = 3
\]
\[
2.2x - 3.3 = -3.3 \Rightarrow 2.2x = 0 \Rightarrow x = 0
\]
- Thus, the solutions are \( x = 3 \) and \( x = 0 \).
Therefore, the answer is:
**x = 0 or x = 3**
9. **Which statement is true?**
- \( -3|2x + 1.2| = -1 \): Has no solution because left-hand side is non-positive.
- \( 3.5|6x – 2| = 3.5 \): One solution occurs when the absolute value equals 1.
- \( 5|–3.1x + 6.9| = –3.5 \): Has no solution because cannot be negative.
- \( -0.3|3 + 8x| = 0.9 \): Has no solution.
The only true statement is:
**The equation –3|2x + 1.2| = –1 has no solution.**
10. **Which number line represents the solutions to |–2x| = 4?**
- The solutions are:
\[
-2x = 4 \Rightarrow x = -2
\]
\[
-2x = -4 \Rightarrow x = 2
\]
- Look for the number line showing dots at **-2 and 2**.
To summarize all answers:
1. She is not correct, because there are no solutions.
2. |–6x + 3| = 0
3. The number line with dots at -2 and -6.
4. x = -4
5. |x – 323| = 50
6. 492 meters and 508 meters
7. The number line with dots at 4 and 6.
8. x = 0 or x = 3
9. The equation –3|2x + 1.2| = –1 has no solution.
10. The number line with dots at -2 and 2.
1. **Misha found that the equation –|2x – 10| – 1 = 2 had two possible solutions: x = 3.5 and x = –6.5. Which explains whether or not her solutions are correct?**
To check Misha's solutions, we can substitute each value back into the original equation:
- For \( x = 3.5 \):
\[
-|2(3.5) - 10| - 1 = -|7 - 10| - 1 = -|-3| - 1 = -3 - 1 = -4 \neq 2
\]
- For \( x = -6.5 \):
\[
-|2(-6.5) - 10| - 1 = -|-13 - 10| - 1 = -|-23| - 1 = -23 - 1 = -24 \neq 2
\]
Both solutions are incorrect. Therefore, the answer is:
**She is not correct, because there are no solutions.**
2. **Which equation has only one solution?**
- \( |x – 5| = -1 \): No solution, as absolute values cannot equal a negative number.
- \( |–6 – 2x| = 8 \): Two solutions.
- \( |5x + 10| = 10 \): Two solutions.
- \( |–6x + 3| = 0 \): One solution (only when the expression inside the absolute value is 0).
Therefore, the answer is:
**|–6x + 3| = 0**
3. **Which number line represents the solutions to |x + 4| = 2?**
- The solutions to the equation are:
\[
x + 4 = 2 \Rightarrow x = -2 \quad \text{and} \quad x + 4 = -2 \Rightarrow x = -6
\]
- Thus, the solutions are \( x = -2 \) and \( x = -6 \).
The number line that shows dots at **-6 and -2** would be the correct one.
4. **Manuela solved the equation 3−2|0.5x+1.5|=2 for one solution. What is the other solution to the equation?**
- From her solution:
\[
|0.5x + 1.5| = 0.5
\]
- This will yield:
\[
0.5x + 1.5 = 0.5 \Rightarrow 0.5x = -1 \Rightarrow x = -2
\]
\[
0.5x + 1.5 = -0.5 \Rightarrow 0.5x = -2 \Rightarrow x = -4
\]
- Thus, the other solution is \( x = -4 \).
Therefore, the answer is:
**x = -4**
5. **For water to be a liquid, its temperature must be within 50 Kelvin of 323 Kelvin. Which equation can be used to determine the minimum and maximum temperatures?**
- The equation describes the distance from 323 to be ±50:
\[
|x - 323| = 50
\]
Therefore, the answer is:
**|x – 323| = 50**
6. **Morgan's dog on an 8-meter-long leash. What are the minimum and maximum distances?**
- The equation is \( |x - 500| = 8 \).
- This gives:
\[
x - 500 = 8 \Rightarrow x = 508
\]
\[
x - 500 = -8 \Rightarrow x = 492
\]
- Thus, the minimum distance is 492 meters and the maximum distance is 508 meters.
Therefore, the answer is:
**492 meters and 508 meters**
7. **Which number line represents the solutions to |x – 5| = 1?**
- The solutions are:
\[
x - 5 = 1 \Rightarrow x = 6
\]
\[
x - 5 = -1 \Rightarrow x = 4
\]
- Therefore, the number line showing dots at **4 and 6** would be the correct choice.
8. **What is the solution to –2|2.2x – 3.3| = –6.6?**
- Simplifying gives us:
\[
|2.2x - 3.3| = 3.3
\]
- This yields:
\[
2.2x - 3.3 = 3.3 \Rightarrow 2.2x = 6.6 \Rightarrow x = 3
\]
\[
2.2x - 3.3 = -3.3 \Rightarrow 2.2x = 0 \Rightarrow x = 0
\]
- Thus, the solutions are \( x = 3 \) and \( x = 0 \).
Therefore, the answer is:
**x = 0 or x = 3**
9. **Which statement is true?**
- \( -3|2x + 1.2| = -1 \): Has no solution because left-hand side is non-positive.
- \( 3.5|6x – 2| = 3.5 \): One solution occurs when the absolute value equals 1.
- \( 5|–3.1x + 6.9| = –3.5 \): Has no solution because cannot be negative.
- \( -0.3|3 + 8x| = 0.9 \): Has no solution.
The only true statement is:
**The equation –3|2x + 1.2| = –1 has no solution.**
10. **Which number line represents the solutions to |–2x| = 4?**
- The solutions are:
\[
-2x = 4 \Rightarrow x = -2
\]
\[
-2x = -4 \Rightarrow x = 2
\]
- Look for the number line showing dots at **-2 and 2**.
To summarize all answers:
1. She is not correct, because there are no solutions.
2. |–6x + 3| = 0
3. The number line with dots at -2 and -6.
4. x = -4
5. |x – 323| = 50
6. 492 meters and 508 meters
7. The number line with dots at 4 and 6.
8. x = 0 or x = 3
9. The equation –3|2x + 1.2| = –1 has no solution.
10. The number line with dots at -2 and 2.
Answered by
GPT-4o mini
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