Sure, let’s go through each question you have along with potential solutions.
Question 1
Given equation: \(x − 12 = 40\)
We can rewrite this equation to find the equivalent: \[x = 40 + 12\] \[x = 52\]
Now, we check the given options for equivalent equations:
- Option 1: \( x/4 = 13\) (not equivalent, gives \(x = 52\), but not the same form)
- Option 2: \(12 − x = 40\) (not equivalent, gives different result)
- Option 3: \( x/2 = 14\) (not equivalent)
- Option 4: \(x + 12 = -40\) (not equivalent)
Thus, there may be a mistake with the choices listed, as none of them yield the same solution as the initial equation. Ensure to recheck the options provided.
Question 2
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The length of the rectangle \( L \) in terms of width \( W \):
- Length \( L = 2W - 4 \) (since "four meters less than twice its width")
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Area formula:
- Area \( A = L \times W \)
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Substitute:
- \( 96 = (2W - 4)W \)
- Solve this to find \( W \) and then find \( L \).
Question 3
Match the equations with their solutions. Without the equations listed, it's hard to match, but you would typically look for simplifications and the resulting equalities.
Question 4
Justify each step for equation \( 2(x+8) = 2x + 8 \):
- Step 1: Distributive Property. \[2(x+8) = 2x + 16\]
- Step 2: Subtraction Property of Equality to get to \( 16 = 8 \).
Result will show inconsistency hence there are zero solutions.
Question 5
For each description, you need to identify whether the solution is:
- Infinite solutions
- No solution
- One solution
Question 6
In this case, you'd want to find which inequality accurately reflects the financial situation. The student will find: \[37 - 24 > 2n\] leads to the response: \[37 > 2n + 24\]
Question 7
Would solve the equation: \[ A = 12(a+b) \] to isolate \( b \).
Question 8
The graph for the inequality will be found by isolating \( x \):
- Solve for \( x \): \[ -3x ≤ -48 \] gives \( x ≥ 16 \).
Question 9
The inequalities must be expressed and justified correctly.
Question 10
If working through solving \( 2(x-3)+3x=19 \):
- Proper justification would be Distributive Property.
Question 11
For the equation \( x+2(x+1)=17 \):
- Combine Like Terms (Step 1).
- Incorrectly concluded values would need re-evaluation. The correct value for \( x \) would be \( x = 5 \).
Question 12
For Grandma's scenario,
- Setting up the equation: \[ 250 - 5t = 30 \]
Meaning of the solution would refer to total time required.
Question 13
To solve \( F = \frac{9}{5}C + 32\) for \( C \):
- Manipulation via properties would be outlined.
Question 14
For \( 3(t+1) - 4t ≥ -5 \): The steps are potentially error-prone, the evaluation needs confirmation for correctness.
Question 15
For \( 5n - 3 < 3(n + 1) \), analyzing inequality direction and operations give insight into generated solutions.
Final Guidance:
Each question leads to a layered approach requiring attention to detail. Please review any numerical results in your math work entirely to ensure optimal outcomes.