Let's go through each question step by step and provide the correct responses:
Question 7 (Inequality)
The scenario involves a student wanting to buy a calculator and some notebooks within a limited budget. The inequality can be formulated as follows:
- The cost of the calculator is $24.
- The cost of each notebook is $2.
- The total budget is $37.
The inequality representing the scenario would be:
\[
37 \geq 2n + 24
\]
The correct response is:
37≥2n+24
Question 8 (Equation for b)
To solve for \(b\) in the equation \(A = 12(a + b)\), we can rearrange the equation:
\[
A = 12a + 12b
\]
Now, isolating \(b\):
\[
12b = A - 12a \implies b = \frac{A - 12a}{12}
\]
The options provided have some mismatch, but a likely intended option is:
b = 12A − a.
(Please note that this doesn't match perfectly with the direct rearrangement found above but appears to be the closest response.)
Question 9 (Graphing Inequality)
To solve and graph \(-3x + 1 \leq -47\):
- Subtract 1 from both sides:
\[-3x \leq -48\]
- Divide by -3 and switch the inequality:
\[x \geq 16\]
This means we would shade to the right of 16 (including 16).
Question 10 (Student Mistake)
The student's claim is based on the inequality:
\(-4 < x\). The mistake could be:
The student did x is less than -4, when the variable is on the other side; -4 is less than x, so x is greater than -4.
Question 11 (Property Justification)
In the steps provided for solving:
\[
5x - 6 = 19 \quad (Step 3)
\]
To get to
\[
5x = 25 \quad (Step 4)
\]
The property used is:
Addition Property of Equality.
Question 12 (Mistakes in Solving Equation)
Property used to arrive at Step 1:
- The response is: Distributive Property (used to expand \(2(x+1)\)).
Step includes a mistake made from the previous step:
- The response is: Step 4 (incorrectly solved for \(x=45\)), which should have been: Step 3 where the answer was \(3x=15\).
Correct answer for x:
- The response is: x = 5 (correct solving shows \(3x = 15\), thus \(x = 5\)).
Question 13 (Weeding Equation)
Equation:
To find out how many minutes it will take to reduce weeds from 250 to 30:
\[
250 - 5m = 30
\]
Solution:
Reorganizing gives:
\[
5m = 250 - 30 = 220 \implies m = \frac{220}{5} = 44
\]
Meaning of the solution in words:
It will take grandma 44 minutes to remove weeds until only 30 are left.
Question 14 (Celsius to Fahrenheit Conversion)
- First step is to add 32 to both sides. - True.
- First step is to multiply by 9 on each side. - False, that is not the first step.
- The next step from F - 32 = 9/5 C, could be to multiply both sides by 5 - True.
- The next step from F - 32 = 9/5 C, could be to divide both sides by 5. - False, it cannot be the next step.
- Solution:
\[
C = \frac{5}{9}(F - 32)
\] would be the initial solution derived from rearranging correctly.
- To solve this equation for C, you would have to distribute at some point. - False.
Question 15 (Inequality Mistakes)
- Izzie should not have changed the inequality symbol from ≥ to ≤. - True.
- Izzie made no mistakes in solving this inequality. - False.
- Izzie made a mistake going from step 3 to step 4. - True.
- Izzie made a mistake going from step 2 to step 3. - False; her steps were correct until step 3 but incorrect in interpreting results.
- The correct answer should have been t ≤ 8. - False; she solved for t incorrectly.
- Izzie made a mistake going from step 1 to step 2. - False; her first step was correct.
Question 16 (Graph of Inequality)
To solve \(5n - 3 < 3(n + 1)\):
- Distributing gives:
\[5n - 3 < 3n + 3\]
- Reorganizing:
\[5n - 3n < 3 + 3\]
\[2n < 6\]
\[n < 3\]
The graph should indicate \(n\) values shaded to the left of \(3\).
Conclusion
Every question has been addressed with the best possible solution and rationale based on the information provided. Please copy and adapt the responses as needed for your format.
