Question
Parul attempted to solve an inequality but made one or more errors. Her work and the graph she drew are shown below.
Negative 5 x minus 3.5 greater-than 6.5. Negative 5 x greater-than 10. x greater-than negative 50.
A number line going from negative 110 to positive 10. A closed circle is at negative 50. Everything to the right of the circle is shaded.
What errors did Parul make? Select three options.
She added 3.5 to both sides when she should have subtracted.
She should have divided both sides by Negative 5 as her first step.
She divided one side by -5 while multiplying the other side by -5.
She did not change the > symbol to a < symbol.
She used a closed circle instead of an open circle on the number line.
Negative 5 x minus 3.5 greater-than 6.5. Negative 5 x greater-than 10. x greater-than negative 50.
A number line going from negative 110 to positive 10. A closed circle is at negative 50. Everything to the right of the circle is shaded.
What errors did Parul make? Select three options.
She added 3.5 to both sides when she should have subtracted.
She should have divided both sides by Negative 5 as her first step.
She divided one side by -5 while multiplying the other side by -5.
She did not change the > symbol to a < symbol.
She used a closed circle instead of an open circle on the number line.
Answers
GPT-4o mini
To identify the errors Parul made in solving the inequality \(-5x - 3.5 > 6.5\), let's go through the steps systematically and evaluate the options provided.
1. **Start with the original inequality:**
\(-5x - 3.5 > 6.5\)
2. **Isolate the variable term:**
To isolate \(-5x\), we should add \(3.5\) to both sides:
\(-5x > 6.5 + 3.5\)
\(-5x > 10\)
3. **Now, divide both sides by \(-5\):**
When we divide by a negative number, we must flip the inequality sign:
\(x < \frac{10}{-5}\)
\(x < -2\)
The correct solution is \(x < -2\). However, in Parul's work, she incorrectly concluded that \(x > -50\), which is incorrect.
Now let's evaluate the options:
1. **She added 3.5 to both sides when she should have subtracted.**
- This option is incorrect since she correctly added \(3.5\) after isolating the -5x term.
2. **She should have divided both sides by Negative 5 as her first step.**
- This is incorrect because we cannot divide by \(-5\) before isolating the term.
3. **She divided one side by -5 while multiplying the other side by -5.**
- This is a mistake since she did not perform the operations correctly when dividing both sides. Rather, she divided one side and then incorrectly manipulated the inequality.
4. **She did not change the > symbol to a < symbol.**
- This is true. After dividing by a negative number, she should have flipped the sign, but she did not.
5. **She used a closed circle instead of an open circle on the number line.**
- This is a confusion point; she should use an open circle for 'less than' inequalities since \(-2\) is not included in the solution.
Given the analysis, the three correct errors Parul made are:
- She divided one side by -5 while multiplying the other side by -5.
- She did not change the > symbol to a < symbol.
- She used a closed circle instead of an open circle on the number line.
So the selected options indicate the mistakes Parul made clearly and accurately.
1. **Start with the original inequality:**
\(-5x - 3.5 > 6.5\)
2. **Isolate the variable term:**
To isolate \(-5x\), we should add \(3.5\) to both sides:
\(-5x > 6.5 + 3.5\)
\(-5x > 10\)
3. **Now, divide both sides by \(-5\):**
When we divide by a negative number, we must flip the inequality sign:
\(x < \frac{10}{-5}\)
\(x < -2\)
The correct solution is \(x < -2\). However, in Parul's work, she incorrectly concluded that \(x > -50\), which is incorrect.
Now let's evaluate the options:
1. **She added 3.5 to both sides when she should have subtracted.**
- This option is incorrect since she correctly added \(3.5\) after isolating the -5x term.
2. **She should have divided both sides by Negative 5 as her first step.**
- This is incorrect because we cannot divide by \(-5\) before isolating the term.
3. **She divided one side by -5 while multiplying the other side by -5.**
- This is a mistake since she did not perform the operations correctly when dividing both sides. Rather, she divided one side and then incorrectly manipulated the inequality.
4. **She did not change the > symbol to a < symbol.**
- This is true. After dividing by a negative number, she should have flipped the sign, but she did not.
5. **She used a closed circle instead of an open circle on the number line.**
- This is a confusion point; she should use an open circle for 'less than' inequalities since \(-2\) is not included in the solution.
Given the analysis, the three correct errors Parul made are:
- She divided one side by -5 while multiplying the other side by -5.
- She did not change the > symbol to a < symbol.
- She used a closed circle instead of an open circle on the number line.
So the selected options indicate the mistakes Parul made clearly and accurately.